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* 

WELLS'   MATHEMATICAL   SERIES. 


Academic  Arithmetic. 

Academic  Algebra. 

Higher  Algebra. 

University  Algebra. 

College  Algebra. 

Plane  Geometry. 

Solid  Geometry. 

Plane  and  Solid  Geometry. 

Plane  and  Solid  Geometry.    Revised. 

Plane  and  Spherical  Trigonometry. 

Plane  Trigonometry. 

Essentials  of  Trigonometry. 

Logarithms  (flexible  covers). 

Elementary  Treatise  on  Logarithms. 


Special  Catalogue  and  Terms  on  application. 


*- 


AN 


Academic   Arithmetic 


ACADEMIES,  HIGH  AND  COMMERCIAL  SCHOOLS. 


BY 


WEBSTER  WELLS,   S.B., 

PROFESSOR  OF   MATHEMATICS   IN   THE   MASSACHUSETTS 
INSTITUTE   OF  TECHNOLOGY. 


LEACH,    SHEWELL,   &   SANBOEK 
BOSTON.     NEW  YORK.     CHICAGO. 


COPYBIGHT,  1893. 

By  WEBSTER  WELLS. 


NarbJootJ  3Pre88 : 

J,  S.  Gushing  &  Co.  —  Berwick  &  Smith. 

Boston,  Mass.,  U.S.A. 


PREFACE 


rr^HE  present  work  is  intended  to  furnish,  a  thorough. 
-*-  course  in  all  those  portions  of  Arithmetic  which  are 
required  for  admission  to  any  college  or  scientific  school. 

The  pupil  is  assumed  to  have  already  studied  the  more 
elementary  parts  of  the  subject  in  a  text-book  of  lower 
grade ;  and  only  enough  examples  are  given  in  the  earlier 
chapters  to  afford  material  for  a  review. 

Great  pains  have  been  taken,  in  the  selection  of  exam- 
ples and  problems,  to  illustrate  every  important  arithmet- 
ical process;  and  in  Chapter  XXV.  there  will  be  found  a 
set  of  miscellaneous  problems  of  somewhat  greater  diffi- 
culty than  those  in  the  preceding  chapters,  furnishiug  a 
complete  review  of  the  entire  subject. 

The  chapters  on  the  Metric  System  have  been  arranged 
in  such  a  way  that  they  may  be  taken  after  the  other 
portions  of  the  work  have  been  studied,  or  omitted  alto- 
gether, at  the  option  of  the  teacher.  No  examples  involv- 
ing a  knowledge  of  the  Metric  Systepi  are  given,  except 
in  Chapter  XIII.,  and  Arts.  256,  257,  and  378. 

The  Appendix  contains  topics  of  minor  importance  to 
the  majority  of  pupils,  but  still  liable  to  be  called  for  in 
college  entrance  examinations. 

WEBSTER   WELLS. 

Mass.  Institute  of 
Technology,  1893. 


iii 


'  O  O 


IG 


CONTENTS. 


PAGB 

I.   Notation  and  Numeration 1 

The  Arabic  System  of  Notation 1 

The  Koman  System  of  Notation 5 

II.   Addition 6 

III.  Subtraction 9 

Parentheses 11 

IV.  Multiplication 12 

V.   Division 18 

VI.   Factoring 24 

Casting  out  Nines 29 

Casting  out  Elevens ■ 32 

VII.   Greatest  Common  Divisor 33 

VIII.   Least  Common  Multiple 38 

IX.   Fractions 43 

Reduction  of  Fractions 44 

Addition  of  Fractions 51 

Subtraction  of  Fractions 53 

Multiplication  of  Fractions 55 

Division  of  Fractions 58 

Complex  Fractions 61 

Greatest  Common  Divisor  of  Fractions    ....  65 

Least  Common  Multiple  of  Fractions       ....  66 

Miscellaneous  Examples 66 

Problems 68 

V 


vi  CONTENTS. 

PAGE 

X.   Decimals 74 

To  reduce  a  Decimal  to  a  Common  Fraction     .     .  76 

Addition  of  Decimals 76 

Subtraction  of  Decimals 77 

Multiplication  of  Decimals 78 

Division  of  Decimals 81 

To  reduce  a  Common  Fraction  to  a  Decimal     .     .  84 

Circulating  Decimals 86 

To  multiply  or  divide  a  Number  by  an  Aliquot 

Part  of  10,  100,  1000,  etc 88 

Miscellaneous  Examples    .     .  • 89 

United  States  Money 90 

Problems 93 

XI.   Measures .  98 

XII.   Denominate  Numbers      . 103 

Reduction  of  Denominate  Numbers 103 

Addition  of  Denominate  Numbers  , 107 

Subtraction  of  Denominate  Numbers 109 

To  find  the  Difference  in  Time  between  Two  Dates  110 

Multiplication  of  Denominate  Numbers  .     .     .     .  Ill 

Division  of  Denominate  Numbers 112 

To  express  a  Fraction  or  Decimal  of  a  Simple  Num- 
ber in  Lower  Denominations 115 

To  express  a  Denominate  Number  as  a  Fraction  or 

Decimal  of  a  Single  Denomination 116 

To  express  One  Denominate  Number  as  a  Fraction 

or  Decimal  of  Another 117 

Longitude  and  Time 118 

Problems 120 

XIII.  The  Metric  System 125 

Metric  Numbers .     •  130 

Miscellaneous  Problems" 135 

XIV.  Involution  and  Evolution 138 

Square  Root 140 

Cube  Root 144 


CONTENTS. 


Vll 


PAGE 

XV.  Mensuration 151 

Plane  Figures 151 

Solids 160 

Application  of  Mensuration 171 

Capacity  of  Bins,  Tanks,  and  Cisterns     ....  171 

Carpeting  Rooms 172 

Plastering  and  Papering 174 

Board  Measure 176 

Measurement  of  Round  Timber 177 

Specific  Gravity 178 

Geometrical  Explanation  of  Square  and  Cube  Root  179 
Problems  in  Mensuration  Involving  the  Metric  Sys- 
tem      181 

XVI.   Ratio  and  Pkoportion 191 

Proportion 192 

Properties  of  Proportions 193 

Problems 194 

Compound  Proportion 197 

Partitive  Proportion 201 

Similar  Surfaces  and  Solids 203 

XVII.   Partnership 206 

Simple  Partnership 206 

Compound  Partnership 207 

XVIII.   Percentage 210 

Applications  of  Percentage 221 

Trade  Discount 221 

Commission  and  Brokerage 224 

Insurance 227 

Taxes 229 

Duties 231 

XIX.   Interest 234 

Simple  Interest 234 

The  Six  Per  Cent  Method 236 

Exact  Interest 245 

Promissory  Notes 246 

Partial  Payments 248 

Compound  Interest 252 

Annual  Interest 258 


viii  CONTENTS. 

PAQB 

XX.   Discount 259 

True  Discount 259 

Bank  Discount 260 

XXI.   Exchange 265 

Domestic  Exchange 267 

Foreign  Exchange 270 

XXII.   Equation  of  Payments 274 

Average  of  Accounts 276 

The  Interest  Method 282 

XXIII.  Stocks  and  Bonds 284 

XXIV.  Progressions 292 

Arithmetical  Progression 292 

Geometrical  Progression 295 

Compound  Interest 298 

Annuities 299 

XXV.   Miscellaneous  Examples 303 

Miscellaneous  Examples  Involving  the  Metric  Sys- 
tem      317 

Appendix 321 

Measures 321 

Difference  in  Time  between  Two  Dates  ....  323 

Comparison  of  Thermometers 323 

Money  and  Coins 325 

Legal  Rates  of  Interest 328 

Special  State  Kules  for  Partial  Payments     .     .     .  328 

The  Connecticut  Rule 328 

The  New  Hampshire  Rule  for  Partial  Payments  on 
a  Note  or  Other  Obligation,  Drawing  Annual 

Interest 330 

The  Vermont  Rule 331 

To  compute  Interest  on  English  Money  ....  332 

Business  Forms 333 

Savings  Bank  Accounts 334 

Scales  of  Notation 336 


ARITHMETIC. 


I.    NOTATION  AND  NUMERATION. 

1.  Let  us  consider  a  collection  of  things  of  the  same 
kind ;  for  example,  a  collection  of  books. 

In  order  to  find  out  how  many  books  there  are  in  the 
collection,  we  proceed  to  count  them,  as  follows : 

We  take  any  book,  and  call  it  one;  we  then  take  another 
and  call  it  two;  the  next  we  call  three;  the  next /o?^r;  then 
Jive,  six,  seven,  eight,  nine,  ten,  eleven,  twelve,  and  so  on  until 
all  have  been  taken. 

The  expressions  one,  two,  three,  etc.,  used  in  the  above 
process  are  called  Whole  Numbers,  or  Integers. 

2.  Notation  signifies  the  representation  of  numbers  by 
means  of  symbols. 

Numeration  signifies  the  reading  of  numbers  when  ex- 
pressed in  symbols. 

THE  ARABIC  SYSTEM  OF  NOTATION. 

3.  In  the  Arabic  System,  the  numbers  one,  two,  three, 
four.  Jive,  six,  seven,  eight,  and  nine,  are  represented  by  the 
symbols  1,  2,  3,  4,  5,  6,  7,  8,  and  9,  respectively. 

The  symbol  0,  read  zero,  cipher,  or  naught,  when  standing 
by  itself,  signifies  nothing. 

The  above  symbols  are  called  Figures. 

Zero,  and  the  numbers  one,  two,  three,  four,  five,  six,  seven, 
eight,  and  nine,  are  called  Digits. 

1 


2  ARITHMETIC. 

4.  Numbers  greater  than  nine  are  represented  by  writing 
side  by  side  two  or  more  of  the  above  figures. 

The  last  figure  at  the  right  is  said  to  be  in  the  Jirst  place, 
and  denotes  ones  or  units. 

The  figure  just  before  the  last  is  said  to  be  in  the  second 
place,  and  denotes  tens. 

Thus,  10  signifies  one  ten  and  no  ones;  that  is,  ten. 

11  signifies  one  ten  and  orie  07ie;  that  is,  eleven. 

12  signifies  one  ten  and  tivo  ones;  that  is,  twelve. 

In  like  manner,  13, 14, 15, 16, 17, 18,  and  19  represent  the 
next  seven  numbers  in  order ;  that  is,  thirteen,  fourteeyi^  fif- 
teen, sixteen,  seventeen,  eighteen,  and  nineteen. 

Two  tens  are  called  twenty,  and  represented  by  20. 

The  next  nine  numbers  in  order  are  represented  by  21, 
22,  and  so  on  to  29 ;  and  read  twenty-one,  twenty-two,  and  so 
on  to  twenty-nine,  respectively. 

Three  tens  are  called  thirty;  four  tens,  forty;  five  tens, 
fifty;  si:s.  tens,  sixty ;  seven  tens,  seventy ;  eight  tens,  eighty ; 
nine  tens,  ninety;  and  rejjresented  by  30,  40,  50,  60,  70,  80, 
and  90,  respectively ;  and  in  each  case  the  next  nine  num- 
bers in  order  are  named  and  represented  in  a  manner  similar 
to  that  employed  for  numbers  between  20  and  30. 

5.  Ten  tens  are  called  one  hundred. 

A  figure  in  the  third  place  denotes  hundreds. 

Thus,  100  signifies  one  hundred,  no  tens,  and  no  ones;  that 
is,  one  hundred. 

101  signifies  one  hundred,  no  tens,  and  one  one;  and  is  read 
one  hundi^ed  and  one. 

783  signifies  seven  hundreds,  eight  tens,  and  three  ones;  and 
is  read  seven  hundred  and  eighty-three. 

In  like  manner,  any  number  from  one  hundred  to  nine 
hundred  and.  ninety-nine,  may  be  represented  by  three  figures. 

6.  The  following  table  gives  the  signification  of  each  of 
the  first  seven  places  : 


NOTATION  AND  NUMERATION. 


8 


1st;  ones. 

2d;  tens. 

3d;  hundreds. 

4tli ;  tens  of  hundreds,  or  thousands. 

5th ;  tens  of  thousands. 

6th ;  tens  of  tens  of  thousands,  or  hundreds  of  thousands. 

7th ;  tens  of  hundreds  of  thousands,  or  millions. 

Thus,  7306592  signifies  7  millions,  3  hundreds  of  thou- 
sands, no  tens  of  thousands,  6  thousands,  5  hundreds,  9  tens, 
and  2  ones. 

Note.  It  will  be  understood  hereafter  that,  when  the  digits  of  a 
number  are  spoken  of,  we  mean  the  numbers  represented  by  its  figures, 
without  regard  to  the  places  which  they  occupy. 

Thus,  the  digits  of  352  are  3,  5,  and  2,  and  not  300,  50,  and  2. 

7.  The  general  law  exemplified  in  Art.  6  may  be  stated 
as  follows : 

Any  place  signifies  tens  of  the  numbers  signified  by  the  next 
place  to  the  right. 

8.  For  convenience  of  reading,  places  are  divided  into 
periods  of  three  places  each. 

The  first,  second,  and  third  places  form  the  first  or  units^ 
period ;  the  fourth,  fifth,  and  sixth  form  the  second  or  thou- 
sands^ period;  the  third  three,  the  millions'  period;  the 
fourth  three,  the  billions^  period. 

The  table  gives  the  designation  of  each  of  the  first  four- 
teen periods : 


Period. 

Dbsigkation. 

Period. 

Designation. 

First. 

Units. 

Eighth. 

Sextillions. 

Second. 

Thousands. 

Ninth. 

Septillions. 

Third. 

Millions. 

Tenth. 

Octillions. 

Fourth. 

Billions. 

Eleventh. 

Nonillions. 

Fifth. 

Trillions. 

Twelfth. 

Decillions. 

Sixth. 

Quadrillions. 

Thirteenth. 

Undecillions. 

Seventh. 

Quintillions. 

Fourteenth. 

Duodecillions. 

4  ARITHMETIC. 

Thus,  the  number  23,016,797,681  is  read  twenty-three 
billion,  sixteen  million,  seven  hundred  and  ninety-seven 
thousand,  six  hundred  and  eighty-one. 

Note.  The  above  is  the  usual  system  of  numeration.  In  the  Eng- 
lish system,  the  second  six  places  form  the  millions'  period,  the  third 
six  the  billions'  period,  etc.  Thus,  according  to  the  English  system, 
the  number  57,608,351,000,000  would  be  read  fifty-seven  billion,  six 
hundred  and  eight  thousand  three  hundred  and  fifty-one  million. 

EXAMPLES. 
9.  Read  the  following  numbers  : 


1. 

2705618. 

6. 

144710325046728. 

2. 

6520741869. 

7. 

9080600713256. 

3. 

101294705. 

8. 

280115769001342. 

4. 

78220615437. 

9. 

294007386045. 

6. 

35400986. 

10. 

48520010964700. 

Write  the  following  numbers  in  figures  : 

11.  One  million,  three  hundred  and  twenty-five  thousand, 
seven  hundred  and  twenty-six. 

12.  Five  billion,  seven  hundred  and  eighty  thousand,  two 
hundred  and  five. 

13.  Seventy-nine  million,  one  hundred  and  sixty  thou- 
sand, and  four. 

14.  Sixty-five  billion,  eight  hundred  and  three  million, 
one  hundred  and  eighty-nine  thousand,  four  hundred  and 
fifty. 

15.  Three  hundred  and  fifty -six  million,  eighty-one  thou- 
sand, six  hundred  and  twelve. 

16.  Two  hundred  and  thirty-five  billion,  nine  hundred 
and  twenty-seven. 

17.  Eighty-five  billion,  two  hundred  and  seventy-seven 
million,  six  thousand,  one  hundred. 

18.  Four  trillion,  one  hundred  and  sixty  billion,  twenty- 
seven  million,  one  hundred  and  sixteen  thousand,  and  eighty- 
ihree. 


NOTATION  AND  NUMERATION.  5 

19.  Seven  quadrillion,  eight  hundred  and  twenty-five 
trillion,  four  hundred  and  sixy-three  million,  four  hundred 
and  forty-five. 

20.  Nine  hundred  quintillion,  five  hundred  and  twenty 
billion,  seventy  thousand,  three  hundred  and  fourteen. 


THE  ROMAN  SYSTEM  OF  NOTATION. 

10.  In  the  Roman  System,  the  numbers  one,  Jive,  ten,  fifty, 
one  hundred,  five  hundred,  and  one  thousand  are  represented 
by  the  letters  I,  V,  X,  L,  C,  D,  and  M,  respectively. 

Numbers  other  than  the  above  are  represented  by  writ- 
ing side  by  side  two  or  more  of  the  above  letters. 

When  thus  expressed,  if  a  letter  is  written  after  another 
letter  of  the  same  or  of  greater  value,  the  sum  of  their  values 
is  represented ;  if  a  letter  is  written  before  another  letter  of 
greater  value,  the  difference  of  their  values  is  represented. 

The  following  table  shows  the  methods  usually  employed 
for  representing  numbers  up  to  five  thousand : 


Roman. 

Arabic. 

Roman. 

Arabic. 

Roman. 

Arabic. 

I 

1 

XV 

15 

xc 

90 

II 

2 

XVI 

16 

c 

100 

III 

3 

XVII 

17 

CI 

101 

IV 

4 

XVIII 

18 

cc 

200 

V 

5 

XIX 

19 

ccc 

300 

VI 

6 

XX 

20 

cccc 

400 

VII 

7 

XXI 

21 

D 

500 

VIII 

8 

XXII 

22 

DC 

600 

IX 

9 

XXX 

30 

DCC 

700 

X 

10 

XL 

40 

DCCC 

800 

XI 

11 

L 

50 

DCCCC 

900 

XII 

12 

LX 

60 

M 

1000 

XIII 

13 

LXX 

70 

MD 

1500 

XIV 

14 

LXXX 

80 

MM 

.2000 

Note.     The  Roman  Method  is  now  rarely  used  except  iornuin- 
hering  chapters  of  books,  hours  on  clock-dials,  etc 


arithmp:tic. 


II.    ADDITION. 

11.  To  Add  two  whole  numbers  is  to  count  upwards  from 
either  of  the  numbers  as  many  units  as  there  are  in  the 
other. 

Thus,  to  add  3  and  5,  we  count  upwards  from  ^Jive  units 
as  follows:  4,  5,  6,  7,  8;  the  result  is  8. 

In  like  manner,  we  may  add  three  or  more  numbers. 

Thus,  to  add  7,  4,  and  8,  we  first  count  upwards  from  7 
four  units,  and  then  count  upwards  8  units  from  the  result. 

The  answer  is  19. 

Note.     The  order  in  which  the  numbers  are  taken  is  immaterial. 

12.  The  result  of  addition  is  called  the  Sum. 

13.  The  symbol  +,  read  ''plus'^  or  ^' and/'  signifies 
addition. 

14.  The  symbol  =  is  read  ^'equals,''  '•'is  equal  to"  or 
''are.'' 

Thus,  3  +  5=8  is  read  "  three  and  five  are  eight." 

15.  1.  Find  the  sum  of  396  and  842. 

396  is  the  same  as  3  hundreds,  9  tens,  and  6  units, 
and  842  the  same  as  8  hundreds,  4  tens,  and  2  units. 
^'*'^  We  write  the  numbers  so  that  units,  tens,   and 


1238   Ans         hundreds  shall  be  in  the  same  vertical  columns. 
The  sum  of  2  units  and  6  units  is-  8  units. 

We  then  write  8  under  the  column  of  units. 

The  sum  of  4  tens  and  9  tens  is  13  tens,  or  1  hundred  and  3  tens. 

We  then  write  3  under  the  column  of  tens,  and  camj  the  1  hundred 
mentally  to  the  column  of  hundreds. 

The  sum  of  1  hundred,  8  hundreds,  and  3  hundreds  is  12  hundreds, 
or  1  thousand  and  2  hundreds. 

We  then  write  2  under  the  column  of  hundreds,  and  1  in  the  thou- 
sands' place  of  the  answer. 

Then  the  required  result  is  1  thousand,  2  hundreds,  3  tens,  and  8 
units,  or  1238. 


ADDITION.  7 

2.  Add  546,  97,  384,  and  780. 

Q_  It  IS  customary  in  practice  to  name  results  only 

when  adding  columns  ;  thus,  in  Ex.  2,  we  say  "  4,  11, 
^84:  17  '' .  write  the  7,  and  carry  1. 

780  Then,  "  1,  9,  17,  26,  30  "  ;  write  the  0  and  carry  3. 

W7,Ans.  Then,  "3,  10,  13,  18." 

From  the  above  examples,  we  derive  the  following 

RULE. 

Write  the  numbers  so  that  units,  tens,  hundreds,  etc,  shall 
he  in  the  same  vertical  columns. 

Add  the  digits  in  the  units'  column. 

If  the  residt  is  less  than  10,  write  it  under  the  column  of 
units;  hut  if  it  is  just  10,  or  more  than  10,  write  the  units  of 
the  sum  under  the  column  of  units,  and  carry  the  tens  mentally 
to  the  next  column  to  the  left. 

Proceed  in  a  similar  manner  with  each  of  the  remaining 
columns.,  and  write  under  the  last  column  its  entire  sum. 

Note.  The  work  may  be  proved  by  performing  the  example  a 
second  time  ;  adding  the  columns  from  top  to  bottom,  instead  of  from 
bottom  to  top. 

Another  method  of  proof  is  to  separate  the  numbers  into  two  parts 
by  a  horizontal  line.  Adding  the  numbers  above  the  line,  then  the 
numbers  below,  and  then  these  two  sums,  the  result  should  agree 
with  that  previously  obtained. 

16.  In  practice,  computers  frequently  add  two  columns 
at  once  ;  thus,  in  the  following  example, 

3527 
8448 
1759 

2872 


16606 

we  should  say  72,  131,  179,  206;  write  06,  and  carry  2; 
then,  2,  30,  47,  131,  166. 


8 


ARITHMETIC. 


17.    Only  quantities  of  the  same  kind  can  he  added. 
Thus,  the  sum  of  7  hooks  and  8  hooks  is  15  books;  but  it 
is  not  possible  to  add  7  hooks  and  8  miles. 


EXAMPLES. 

18. 

Add  the  following 

1. 

2. 

3. 

4. 

5. 

1789 

5403 

4529 

7854 

1827 

6543 

786 

7992 

6215 

4329 

2177 

9230 

467 

9448 

25070 

915 

1157 

8920 

4007 

6118 

6783 

898 

3508 

651 

2522 

325 

7526 
7. 

2463 
8. 

5869 
9. 

3909 

6. 

10. 

79856 

45340 

43765 

59864 

77167 

35117 

10087 

89140 

86723 

63489 

32949 

76322 

35174 

48213 

68791 

18817 

36450 

64385 

54876 

74153 

85622 

78809 

79160 

.  83538 

84375 

16304 

39713 

23099 

7176^ 

97561 

III.    SUBTRACTION. 

19.  To  Subtract  one  whole  number  from  another  is  to 
count  downwards  from  the  second  number  as  many  units  as 
there  are  in  the  first. 

Thus,  to  subtract  5  from  8,  we  count  downwards  from  8 
Jive  units,  as  follows :  7,  6,  5,  4,  3 ;  the  result  is  3. 

20.  The  number  to  be  subtracted  is  called  the  Subtrahend. 
The  number  from  which  the  subtrahend  is  to  be  sub- 
tracted is  called  the  Minuend. 

The  result  is  called  the  Remainder  or  Difference. 

21.  The  symbol  — ,  read  '^minus'^  or  "less,"  signifies 
subtraction. 

Thus,  8  —  5  =  3  is  read  "  eight  less  five  are  three." 

22.  It  is  evident  from  Art.  19  that,  if  we  count  upwards 
from  the  Eemainder  as  many  units  as  there  are  in  the  Sub- 
trahend, we  shall  obtain  the  Minuend. 

That  is,  the  Minuend  is  the  sum  of  the  Subtrahend  and 
Eemainder. 

23.  1.  Subtract  483  from  758. 

758  is  the  same  as  7  hundreds,  5  tens,  and  8  units, 
and  483  the  same  as  4  hundreds,  8  tens,  and  3  units. 
483  ^j^Q  write  the  subtrahend  under  the  minuend  so  that 

275  Ans      ^^i^s,  tens,  and  hundreds  shall  be  in  the  same  vertical 
columns. 
3  units  from  8  units  leave  5  units. 

We  cannot  take  8  tens  from  5  tens  ;  but  we  can  take  1  hundred,  or 
10  tens,  from  the  7  hundreds  of  the  minuend,  leaving  6  hundreds  ;  and 
adding  the  10  tens  to  the  5  tens,  we  have  15  tens  ;  then  8  tens  from  15 
tens  leave  7  tens. 

Finally,  4  hundreds  from  6  hundreds  leave  2  hundreds. 

Then  the  required  result  is  2  hundreds,  7  tens,  and  5  units,  or  275. 


10  ARITHMETIC. 

Now  instead  of  taking  1  hundred  from  the  7  hundreds  of 
the  minuend,  in  the  above  example,  we  may  get  the  same 
result  as  follows : 

Adding  1  hundred  to  the  4  hundreds  of  the  subtrahend, 
we  have  5  hundreds. 

Then,  5  hundreds  from  7  hundreds  leave  2  hundreds. 

This  second  method  is  far  preferable  to  the  first. 

From  the  second  method  of  the  above  example,  we  derive 
the  following 

RULE. 

Write  the  subtrahend  under  the  minuend,  so  that  units,  tens, 
hundreds,  etc.,  shall  he  in  the  same  vertical  columns. 

Subtract  the  right-hand  digit  of  the  subtrahend  from  the 
digit  above  it,  and  write  the  result  under  the  column  of  units. 

If  the  right-hand  digit  of  the  subtrahend  is  greater  than 
the  digit  above  it,  increase  the  latter  by  10  before  subtracting; 
and  add  1  mentally  to  the  digit  in  the  tens'  place  of  the  sub- 
trahend. 

Proceed  in  a  similar  manner  with  each  of  the  remaining 
digits  of  the  subtrahend  in  order. 

Note  1.  If  the  minuend  has  more  places  than  the  subtrahend,  we 
may  make  the  nuniber  of  places  in  the  latter  the  same  as  in  the 
.former  by  mentally  supplying  ciphers  in  the  missing  places. 

2.  Subtract  3728  from  571000. 

571000  In  this  case  we  say,  "8  from  10  leaves  2;  3 

3728  from  10  leaves  7  ;  8  from  10  leaves  2  ;  4  from  11 

567272,  Ans.       l^^v^s  7  ;  1  from  7  leaves  6." 

Note  2.  Since  the  minuend  is  the  sum  of  the  subtrahend  and 
remainder  (Art.  22),  the  work  may  be  proved  by  adding  the  subtra- 
hend to  the  remainder ;  the  result  should  equal  the  minuend. 

24.    Only  quantities  of  the  same  kind  can  be  subtracted. 
Thus,  16  books  less  7  books  are  9  books;  but  it  is  not 
possible  to  subtract  7  books  from  16  miles. 


SUBTRACTION. 


11 


EXAMPLES. 

25.  Subtract  the  following : 

1. 

2. 

3. 

4. 

5. 

6372 

5034 

8000 

9037 

48609 

2177 

786 

1256 

3409 

9085 

6. 

7. 

8. 

9. 

10. 

75816 

40709 

58000 

88713 

64751 

38912 

36090 

49374 

73536 

10968 

PARENTHESES. 

26.  A  Parenthesis,  (  ),  signifies  that  the  numbers  enclosed 
by  it  are  to  be  taken  collectively. 

Thus,  17— (8  +  4)  signifies  that  8  and  4  are  to  be  added 
together,  and  their  sum  subtracted  from  17. 

The  Vinculum, ,  has  the  same  force  as  a  parenthesis. 

Thus,  33  —  11  +  19  —  5  signifies  that  11  is  to  be  sub- 
tracted from  33,  then  5  from  19,  and  the  second  result 
added  to  the  first. 


EXAMPLES. 
Eind  the  values  of  the  following : 
1.   32 -(13 +  7).  4.  122 -(97 -69). 


2.  50-17  +  35-16.  5.  171-119  +  137-88.. 

3.  (45 +  18) -(22 +  9).      6.   (926 -265) -(284 +  198). 


7.    (823 -486) -75^ -515. 


8.    (132 -74) +  115 +  97 -(183 -66). 


9.    1088  -  905  -  323  -  479  -  741  -  262. 

Note.     Brackets,   [  ],  and  Braces,  {  },  liave  the  same  force  as 
parentheses. 


12  ARITHMETIC. 


IV.    MULTIPLICATION. 

27.  To  Multiply  one  whole  number  by  another  is  to  take 
the  first  number  as  many  times  as  there  are  units  in  the 
second. 

Thus,  to  multiply  3  by  5,  we  take  3  five  times,  as  follows : 
3  +  3  +  3  +  3  +  3;  the  result  is  15. 

28.  The  number  taken  is  called  the  Multiplicand. 

The  number  which  shows  how  many  times  the  multipli- 
cand is  taken,  is  called  the  Multiplier. 

The  result  of  multiplication  is  called  the  Product. 

29.  The  symbol  x,  read  ''times/'  signifies  multiplication. 
Thus,  3  X  5  =  15  is  read  "  three  times  five  are  fifteen." 

30.  To  multiply  5  by  3,  we  take  5  three  times,  as  follows : 
5  +  5  +  5 ;  the  result  is  15. 

That  is,  3  X  5  is  equal  to  5  x  3. 

It  is  evident  from  this  that,  in  finding  the  product  of  two 
numbers,  either  may  be  regarded  as  the  multiplicand,  and 
the  other  as  the  multiplier. 

31.  To  multiply  together  three  or  more  numbers,  we 
multiply  the  first  number  by  the  second,  the  product  by 
the  third  number,  and  so  on  until  all  have  been  taken. 

It  is  evident,  as  in  Art.  30,  that  the  order  in  which  the 
numbers  are  multiplied  is  immaterial. 

32.  If  any  number  of  tilings  of  the  same  kind  be  multi- 
plied by  a  whole  number,  the  product  will  be  things  of  the 
same  kind  as  the  multiplicand. 

Thus,  6  times  7  hooks  are  42  books.' 

But  it  is  not  possible  to  take  6  books  times  7  books,  nor 
to  multiply  6  by  7  books. 

33.  The  products  of  the  numbers  from  1  to  12  inclusive, 
taken  two  and  two,  are  given  in  the  following  table. 


MULTIPLICATION. 


13 


MULTIPLICATION  TABLE. 


1 

2 
3 
4 
5 

6 
7 
8 
9 
10 

2 

4 

6 

8 
10 
12 
14 
16 
18 
20 

3 
6 
9 
12 
15 
18 
21 
24 
27 
30 

4 
8 
12 
16 
20 
24 
28 
32 
36 
40 

5 

10 
15 
20 
25 
30 
35 
40 
45 
60 

6 
12 
18 
24 
30 
36 
42 
48 
54 
60 

7 

14 
21 
28 
35 
42 
49 
56 
63 
70 

8 
16 
24 
32 
40 
48 
56 
64 
72 
80 

9 
18 
27 
36 
45 
54 
63 
72 
81 
90 

10 
20 
30 
40 
50 
60 
70 
80 
90 
100 

11 
22 
33 
44 

55 
66 

77 
88 
99 
110 

12 
24 
36 

48 
60 
72 
84 
96 
108 
120 

11 
12 

22 
24 

33 

36 

44 

48 

55 
60 

66 

72 

77 
84 

88 
96 

99 
108 

110 
120 

121 
132 

132 

144 

The  arrangement  of  tlie  table  is  as  follows : 
To  find  the  product  of  7  and  9,  look  in  the  left-hand  ver- 
tical column  for  7 ;  then  the  product  required  will  be  found 
in  the  corresponding  horizontal  line  in  the  column  headed 
9 ;  the  result  found  is  63. 


34.   1.   Multiply  462  by  7. 
Multiplicand,  462  462  is  the  same  as  4  hundreds,  6 

Multiplier,  7  ^ens,  and  2  units. 

T>     J      i.  oooT      A  We  write  the  multiplier  under  the 

Product,  3234,  Ans.         .^  ,  _  .  ^.         .;.  ,.       , 

'  '  units'  figure  of  the  multiplicand. 

7  times  2  units  are  14  units,  or  1  ten  and  4  units. 

We  then  write  4  under  the'  column  of  units. 

7  times  6  tens  are  42  tens ;  adding  to  this  the  1  ten  reserved  from 

the  14  units,  we  have  43  tens,  or  4  hundreds  and  3  tens. 

We  then  write  3  under  the  column  of  tens. 


14  ARITHMETIC. 

7  times  4  hundreds  are  28  hundreds ;  adding  to  thi^  the  4  hundreds 
reserved  from  the  43  tens,  we  have  32  hundreds,  or  3  thousands  and 
2  hundreds. 

Writing  2  under  the  column  of  hundreds,  and  3  in  the  thousands' 
place  of  the  product,  the  required  result  is  3234. 

It  is  customary  to  use  the  following  words  only  in  ex- 
plaining the  above  process : 

7  times  2  are  14 ;  write  the  4,  and  "carry"  1 ;  7  times  6  are  42, 
and  1  are  43  ;  write  the  3,  and  carry  4  ;  7  times  4  are  28,  and  4  are  32. 

2.   Multiply  743  by  685. 

Multiplicand,  743  We  multiply  743  first  by. 5 

Multiplier  685  units,  then  by  8  tens,  and  finally 

1st  partial' product,  3715  ^^  ^  hundreds,  and  add  the 

oj         i.-  1          3     J  ^f^AAf^  partial  products. 

2d  partial  product,  59440  743  ^^^^^  ^  ^^^^  ^^^  37^5 

3d  partial  product,  445800  units. 

Product,  508955,  Ans.      743  times  8  tens  are  5944 

tens,  or  59440. 

743  times  6  hundreds  are  4458  hundreds,  or  445800. 

Adding  3715,  59440,  and  445800,  the  required  result  is  508955. 

It  is  customary  to  arrange  the  written  work  as  follows : 

743 

685 


3715 
5944 
4458 

508955,  Ans. 

From  the  above  example,  we  derive  the  following 

RULE. 

WHte  the  multiplier  under  the  multiplicand,  so  that  units, 
tens,  hundreds,  etc.,  shall  be  in  the  same  vertical  columns. 

Multiply  the  midtiplicand  by  the  digit  in  the  units^  place  of 
the  multiplier,  and  write  the  result  under  the  multiplier  so  that 
its  right-hand  Jigure  shall  be  under  the  units'  figure  of  the 
multiplier. 


MULTIPLICATION.  15 

Multiply  the  multiplicand  by  the  digit  in  the  tens'  place  of 
the  multiplier,  and  write  the  result  under  the  first  partial 
product  so  that  its  right-hand  figure  shall  be  under  the  tens^ 
figure  of  the  multiplier. 

Proceed  in  a  similar  manner  with  each  of  the  remaining 
digits  of  the  multiplier,  and  add  the  partial  products. 

Note.    The  work  may  be  proved  by  interchanging  the  multiplicand 

and  multiplier. 

If  any  digit  of  the  multiplier  is  0,  the  corresponding 
partial  product  is  0,  and  is  not  expressed  in  the  work. 

3.   Multiply  1725  by  309. 

1725  We  say  in  this  case  :  9  times  1725  are  15525,  which 

309  we  write  so  that  its  right-hand  figure  shall  be  under 

15525  the  9  of  the  multiplier  ;  then,  3  times  1725  are  5175, 

5175  which  we  write  so  that  its  right-hand  figure  shall  be 

533025  A71S      ^^^^^  ^^^  ^  o^  the  multiplier. 

35.  To  Multiply  by  10,  100,  1000,  Etc. 

To  multiply  a  whole  number  by  10,  100,  1000,  etc.,  we 
annex  to  the  multiplicand  as  many  ciphers  as  there  are  in 
the  multiplier. 

Example.     Multiply  356  by  1000. 
Annexing  three  ciphers  to  the  multiplicand,  we  have 
356  X  1000  =  356000,  Ans. 

36.  To  Multiply  by  any  Number  of  Tens,  Hundreds,  Etc. 

Any  number  of  ciphers  at  the  right  of  the  multiplier 
may  be  omitted  during  the  operation  of  multiplication,  and 
annexed  to  the  result. 

1.   Multiply  2734  by  2600. 

2734 

2600  We  say  in  this  case :  6  times  2734  are  16404  ;  2 

16404  times  2734  are  5468;   adding,   and  annexing  two 

5468  ciphers  to  the  result,  the  product  is  7108400. 

7108400,  Ans. 


16  ARITHMETIC. 

In  like  manner,  ciphers  at  the  right  of  the  multiplicand 
may  be  omitted  during  the  operation. 

2.   Multiply  63000  by  580. 

68000 

*^^^  We  say  in  this  case :   8  times  63  are  504  ;   5 

504  times  63   are   315  ;    adding,  and   annexing  four 

315 ciphers  to  the  result,  the  product  is  36540000. 

36540000,  Ans. 

37.  When  two  numbers  are  to  be  multiplied  together, 
circumstances  will  often  determine  which  to  take  as  the 
multiplier. 

In  general,  the  number  having  the  least  number  of  places 
should  be  taken  as  the  multiplier;  thus,  to  multiply  852 
and  27,  we  should  take  27  as  the  multiplier. 

If  one  of  the  numbers  has  two  or  more  digits  alike,  it 
may  be  easier  to  take  it  as  the  multiplier  ;  thus,  to  multiply 
Q&Q  and  329,  it  would  be  easier  to  take  the  former  number 
as  the  multiplier. 

Again,  if  one  of  the  numbers  has  ciphers  or  ones  for  digits, 
it  may  be  shorter  to  take  it  as  the  multiplier;  thus,  to 
multiply  394  and  2001,  it  would  be  shorter  to  take  the  latter 
number  as  the  multiplier. 

38.  Short  Methods  in  Multiplication. 

To  multiply  by  99,  999,  etc,  we  may  proceed  as  follows : 
Example.     Multiply  1652  by  999. 

1652000  Since  999  is  1000  -  1,  we  multiply  1652  by  1000, 

^bo^  and  then  by  1,  and  subtract  the  second  result  from 

1650348,  Ans.     the  first. 

In  like  manner,  we  may  multiply  by  98,  97,  998,  997,  or 
by  any  number  a  little  less  than  100,  1000,  10000,  etc. 

The  same  artifice  may  be  employed  when  the  multiplier 
is  a  little  less  than  200,  300,  2000,  3000,  etc. 

Thus,  to  multiply  867  by  698,  we  multiply  it  by  700,  and 
then  by  2,  and  subtract  the  second  result  from  the  first. 


MULTIPLICATION.  IT* 

EXAMPLES. 
39.  Multiply  the  following: 

1.  873  by  956.  8.  15063  by  9874. 

2.  2600  by  3950.  9.  54189  by  7998. 

3.  487  by  8009.         •  10.  7677  by  4912. 

4.  4067  by  997.  11.  2946  by  5335. 

5.  3476  by  625.  12.  82821  by  7269. 

6.  6872  by  599.  13.  93247  by  2461. 

7.  60507  by  3784.        14.  35895  by  6927. 

15.  Find  the  value  of  (17  +  15)  x  (19  -  3). 

Note.  This  signifies  that  17  and  15  are  to  be  added  together,  then 
3  subtracted  from  19,  and  then  the  first  result  multiplied  by  the 
second.     (Compare  Art.  26.) 

Find  the  values  of  the  following : 

16.  (38  -  15)  X  26.  19.    (92- 36)  x  (88 -29). 

17.  (13  +  21)  X  (32+ 7).        20.    103  x  (186- 115  +  137). 

18.  (77  +  43)  X  (56 -19).     21.   463  -  (35  +  29)  x  5  + 148. 


22.  (391  -  274  -  89)  X  (96 -37). 

23.  (127  -  98)  X  (101  +  66)  -  (103  -  79)  x  (S6  -  47) . 


24.    (856  -  614  -  477)  x  (982  -  378  -f-  249). 


25.    (387  -  35  +  123)  x  (458  -  129-75+48)  +  (713  x  294). 


IB  ARITHMETIC. 


V.    DIVISION. 

40.  To  Divide  one  whole  humber  by  another  is  to  find  a 
number  which,  when  multiplied  by  the  second  number,  will 
produce  the  first. 

Thus,  to  divide  15  by  5  is  to  find  a  number  which  when 
multiplied  by  5,  will  produce  15. 

The  number  is  3 ;  hence,  15  divided  by  5  are  3. 
We  also  say,  "  5  is  contained  in  15  three  times." 

41.  The  number  which  is  divided  is  called  the  Dividend. 
The  number  by  which  the  dividend  is  divided  is  called 

the  Divisor. 

The  result  is  called  the  Quotient. 

42.  It  is  evident  that  the  Dividend  is  the  product  of  the 
Divisor  and  Quotient. 

43.  The  symbol  -f-,  read  ^^  divided  hy,^^  signifies  division. 
Thus,   15  -^  5  =  3  is  read    "  fifteen  divided   by  five  are 

three." 

44.  If  one  number  does  not  exactly  contain  another,  its 
excess  above  the  next  smaller  number  that  does  exactly 
contain  the  second  number  is  called  the  Remainder. 

Thus,  5  is  contained  in  17  three  times,  with  a  remainder  2. 

45.  1.   Divide  852  by  3. 

Divisor,  3)852,  Dividend.  We  write  the  divisor  at  the  left  of 
Quotient,       284,  Ans.  *^^  dividend,  with  a  )  between  them. 

3  is  contained  in  8  hundreds  2 
hundreds  times,  with  a  remainder  of  2  hundreds,  or  20  tens. 

We  then  write  2  under  the  hundreds'  figure  of  the  dividend. 

20  tens  and  5  tens  are  25  tens  ;  3  is  contained  in  25  tens  8  tens 
times,  with  a  remainder  of  1  ten,  or  10  units. 

We  then  write  8  under  the  tens'  figure  of  the  dividend. 

10  units  and  2  units  are  12  units ;  3  is  contained  in  12  units  4  times. 

We  then  write  4  under  the  units'  figure  of  the  dividend. 

The  required  result  is  284. 


DIVISION.  19 

The  following  words  are  used  in  explaining  the  above 
process : 

3  in  8  twice,  with  2  to  carry  ;  3  in  25  eight  times,  with  1  to  carry  ; 
3  in  12  four  times. 

2.   Divide  22236  by  68. 

First  Process.  68  is  contained  in  222  three  times,  with  a  remain- 

68")  22236  ^^^  ^^  ^^  >  ^^  ^^  contained  in  183  twice,  with  a  re- 

^^     A  mainder  of  47  ;  68  is  contained  in  476  seven  times. 

'          *  The  required  result  is  327. 

In  order  to  avoid  the  labor  of  calculating  the  remainders 
mentally,  it  is  customary  to  arrange  the  written  work  as 
follows  : 

Second  Process.  We  say,  68  is  contained  in  222  three  times, 

68)  22236  (327,  Ans.     ^^^h  a  remainder. 

orv^  Multiplying  68  by  3,  the  product  is  204, 

which  we  write  under  the  222  ;  subtracting 

183  204  from  222,  the  remainder  is  18 ;  annex 

135  to  this  the  next  dividend  figure,  3. 

68  is  contained  in  183  twice,  with  a  re- 

4<  6  mainder  ;  multiplying  68  by  2,  the  product 

476  is  136,  which  we  write  under  the  183  ;  sub- 

tracting  136  from  183,  the  remainder  is  47  ; 
annex  to  this  the  last  dividend  figure,  6. 

68  is  contained  in  476  seven  times  ;  multiplying  68  by  7  the  product  is 
476,  which  we  write  under  the  476  ;  subtracting,  there  is  no  remainder. 
Hence,  the  required  result  is  327. 

It  will  be  seen  that  the  second  process  is  essentially  the 
same  as  the  first ;  the  only  difference  being  that,  in  the  first, 
certain  operations  are  performed  mentally,  which  are  written 
out  in  full  in  the  second. 

Note  1.  The  operation  is  called  Short  Division  when  the  remain- 
ders and  partial  products  are  obtained  mentally,  as  in  Ex.  1,  and  the 
first  process  of  Ex.  2  ;  and  Long  Division  when  they  are  written  out 
in  full,  as  in  the  second  process  of  Ex.  2, 

The  method  of  Short  Division  should  always  be  used  when  the 
divisor  is  12  or  less. 


20  ARITHMETIC. 

From  the  second  process  of  Ex.  2,  we  derive  the  following 

RULE  FOR  LONG   DIVISION. 

Write  the  divisor  at  the  left  of  the  dividend. 

Take,  at  the  left  of  the  dividend,  the  smallest  number  of 
digits  that  will  form  a  number  equal  to  or  greater  than  the 
divisor. 

Divide  this  number  by  the  divisor,  arid  write  the  quotient  as 
the  first  digit  of  the  quotient ;  subtract  from  the  number  the 
product  of  the  divisor  by  the  first  digit  of  the  quotient,  and 
annex  to  the  remainder  the  next  figure  of  the  dividend. 

Divide  this  partial  dividend  by  the  divisor,  and  proceed  as 
before;  continuing  the  process  until  all  the  figures  of  the  divi- 
dend have  been  taken. 

Note  2.  If  any  partial  dividend  is  less^than  the  divisor,  write  0 
for  the  corresponding  digit  of  the  quotient,  and  annex  to  the  partial 
dividend  the  next  figure  of  the  dividend. 

Note  3.  If,  on  making  trial  of  any  number  as  a  digit  of  the  quo- 
tient, its  product  by  the  divisor  is  greater  than  the  preceding  partial 
dividend,  the  number  tried  is  too  great,  and  one  less  must  be  substi- 
tuted for  it. 

If  any  remainder  is  equal  to  or  greater  than  the  divisor,  the  digit  of 
the  quotient  last  obtained  is  too  small,  and  one  greater  must  be  sub- 
stituted for  it. 

3.   Divide  80791  by  386. 

In  this  case,  the  smallest  number  of 
(209,  Quotient.        ^^^^^^  ^^  ^he  left  of  the  dividend  that  will 
386)  80791  form  a  number  greater  than  the  divisor, 

772  is  three. 

386  is  contained  in  807  twice,  with  a 
^^^^  remainder  ;  2  times  386  is  772,  which,  sub- 

3474  tracted  from  807,  leaves  35  ;  annex  to  this 

~117,  Remainder.    ^^^  ^^^^  dividend  figure,  9. 

Since  359  is  less  than  the  divisor,  we 
write  0  as  the  second  digit  of  the  quotient,  and  annex  to  359  the  last 
dividend  figure,  1. 

386  is  contained  in  3591  nine  times,  with  a  remainder  ;  .9  times  386 
is  3474,  which,  subtracted  from  3591,  leaves  117. 
Hence,  the  quotient  is  209,  and  the  remainder  117. 


DIVISION.  21 

Note  4.  The  work  may  be  proved  by  multiplying  the  divisor  and 
quotient,  and  adding  the  remainder,  if  any,  to  the  product ;  the  result 
should  equal  the  dividend. 

46.  To  Divide  by  10,  100,  1000,  Etc. 

To  divide  a  whole  number  by  10,  100,  1000,  etc.,  we  cut 
off  from  the  right  of  the  dividend  as  many  digits  as  there 
are  ciphers  in  the  divisor. 

The  result  will  be  the  quotient,  and  the  digits  cut  off  will 
form  the  remainder. 

1.  Divide  360000  by  1000. 

Cutting  ofE  three  digits  from  the  right  of  the  dividend,  we  have 
360000  - 1000  =  360,  Ayis. 

2.  Divide  7298  by  100. 

Cutting  off  two  digits  from  the  right  of  the  dividend,  we  have 
7298  ^  100  =  72,  with  a  remainder  of  98,  Ans. 

47.  To  Divide  by  Any  Number  of  Tens,  Hundreds,  Etc. 

It  is  evident  that,  if  both  dividend  and  divisor  be  divided 
by  the  same  number,  the  value  of  the  quotient  is  not 
changed ;  for  the  new  divisor  is  contained  in  the  new  divi- 
dend just  as  many  times  as  the  old  divisor  is  contained  in 
the  old  dividend. 

It  follows  from  the  above  that  we  may  remove  the  same 
number  of  ciphers  from  the  right  of  both  the  dividend  and 
divisor,  and  find  the  quotient  of  the  resulting  numbers ;  for 
this  is  the  same  as  dividing  both  dividend  and  divisor  by 
the  same  number. 

1.   Divide  400400  by  7700. 

77)4004(52,  Ans. 

ggg  In  this  case,  we  remove  two  ciphers  from 

^f,.  the  right  of  both  the  dividend  and  divisor,  and 

Jg|  divide  4004  by  77. 


22  ARITHMETIC. 

2.   Divide  3358617  by  65000. 

First  Process. 

65000)3358617(51,  Quotient. 
325000 
108617 
65000 
43617,  Remainder. 

The  process  may  be  shortened  by  omitting  the  ciphers  at 
the  right  of  the  divisor,  and  the  same  number  of  digits  at 
the  right  of  the  dividend,  finding  the  quotient  of  the  result- 
ing numbers,  and  annexing  to  the  remainder  the  digits 
omitted  from  the  right  of  the  dividend. 

The  written  work  will  then  stand  as  follows : 
Second  Process. 

65)  3358  I  617  (51,  Quotient. 
325 
108 
65 
43617,  Remainder. 

From  the  above  example,  we  derive  the  following 

RULE. 

If  the  divisor  has,  and  the  dividend  has  not,  ciphers  at  its 
right,  cut  off  the  ciphers  from  the  right  of  the  divisor,  and  the 
same  number  of  digits  from  the  right  of  the  dividend. 

Divide  the  resulting  numbers,  and  annex  to  the  remainder 
the  digits  omitted  from  the  right  of  the  dividend. 

48.  If  the  dividend  has,  and  the  divisor  has  not,  ciphers 
at  its  right,  we  proceed  as  follows  : 

Example.     Divide  476000  by  56. 

56)476000(8500,  Ans.        in  this  case,  56  is  contained  in  4760  85 

448  times ;   annexing  to  this  the  two  ciphers 

280  remaining  at  the  right  of  the  dividend,  the 

280  quotient  is  8500. 


DIVISION.  28 

49.  If  any  number  of  things  of  the  same  kind  be  divided 
by  a  whole  number,  the  quotient  will  be  things  of  the  same 
kind  as  the  dividend. 

Thus,  42  books  divided  by  7  are  6  books. 

Again,  if  any  number  of  things  of  the  same  kind  be 
divided  by  another  number  of  things  of  the  same  kind  as  the 
dividend,  the  quotient  will  be  a  number. 

Thus,  42  books  divided  by  7  books  are  6. 

EXAMPLES. 

50.  Divide  the  following : 

1.  488304  by  12.  8.  53803998  by  10629. 

2.  517668  by  964.  9.  58161020  by  1357. 

3.  63102024  by  6008.  10.  41675206  by  492000. 

4.  24884574  by  49082.  11.  23510372  by  73700. 

6.  68515100  by  69700.  12.   20301129888  by  25376. 

6.  3545000  by  587.  13.   499107840  by  53760. 

7.  83057629  by  10000.  14.   626104565  by  7247. 

15.  Find  the  value  of  (59  -  11)  -  (25  -  17). 

Note.  This  signifies  that  11  is  to  be  taken  from  59,  then  17  from 
25,  and  the  first  result  divided  by  the  second.     (Compare  Art.  26.) 

Find  the  values  of  the  following : 


16.   (78-12-11)  X  (15-7).   20.   (322-106) --(48-21). 


17.   (13  X  5  + 26) -f- 7.  21.   (132-6)  x  (143- 13). 


18.  81  -  (108  ^  9)  +  43.  22.   (297  +  279)  -(6x8). 

19.  (148-15+31) -34.  23.   (33x57) -(442-17). 


24.  (8567  -  12073  -  7988)  ^  (8051  ^  97) , 


24  ARITHMETIC. 


VI.    FACTORING. 

51.  One  whole  number  is  said  to  be  divisible  by  another 
when  the  first  number  can  be  divided  by  the  second  without 
a  remainder. 

In  the  above  case,  the  first  number  is  said  to  be  a  Multiple 
of  the  second,  and  the  second  a  Factor  of  the  first. 
Thus,  15  is  a  multiple  of  5,  and  5  is  a  factor  of  15. 

52.  An  Even  Number  is  one  that  is  divisible  by  2 ;  as  2, 
4,  6,  8,  etc. 

An  Odd  Number  is  one  that  is  not  divisible  by  2 ;  as  1, 
3,  5,  7,  etc. 

53.  A  Prime  Number  is  one  that  is  divisible  only  by  itself 
and  1 ;  as  1,  2,  3,  5,  7,  etc. 

A  Composite  Number  is  one  that  is  divisible  by  other 
numbers  than  itself  and  1 ;  as  4,  6,  8,  9,  10,  etc. 

54.  The  Prime  Factors  of  a  number  are  the  prime  num- 
bers which,  when  multiplied  together,  will  produce  the 
given  number. 

Thus,  the  prime  factors  of  12  are  2,  2,  and  3. 

Note.   It  is  usual  to  exclude  1  in  giving  the  prime  factors  of  a  number. 

To  Factor  a  number  is  to  find  its  prime  factors. 

55.  If  a  number  be  multiplied  by  itself  any  number  of 
times,  the  result  is  called  a  power  of  the  first  number. 

An  exponent  is  a  number  written  at  the  right  of,  and 
above  another,  to  indicate  what  power  of  the  latter  number 
is  to  be  taken ;  thus, 

2^,  read  "2  square^^  or  "2  to  the  2d power,^^  denotes  2x2; 

5^,  read  "  5  cube  "  or  "5  to  the  M  power, ^^  denotes  5x5x5; 

7^  read  "  7  fourth "  or  "  7  to  the  4th  poicer,'^  denotes 
7x7x7x7;  and  so  on. 


FACTORING.  26 

56.   The  following  principles  will  be  found  of  great  use 
in  factoring  numbers : 

Any  number  of  tens  is  divisible  by  2 ;  hence, 

I.  Any  number  is  divisible  by  2  if  its  last  digit  is  0  or  an . 
even  number. 

Thus,  738  is  divisible  by  2,  because  8  is  an  even  number. 

Any  number  of  hundreds  is  divisible  by  4 ;  hence, 

II.  Any  number  is  divisible  by  4  if  the  number  formed  by 
its  last  two  digits  is  divisible  by  4. 

Thus,  3568  is  divisible  by  4,  because  68  is  divisible  by  4. 

Any  number  of  thousands  is  divisible  by  8 ;  hence, 

III.  A7iy  number  is  divisible  by  8  if  the  number  formed  by 
its  last  three  digits  is  divisible  by  8. 

Thus,  47352  is  divisible  by  8,  because  352  is  divisible  by  8. 

Any  number  of  tens  is  divisible  by  5 ;  hence, 

ly.   Any  number  is  divisible  by  5  if  its  last  digit  is  0  or  5. 

Thus,  120  and  8295  are  divisible  by  5. 

Any  number  of  tens  is  divisible  by  10 ;  hence, 

V.  Any  number  is  divisible  by  10  if  its  last  digit  is  0. 
Thus,  3790  is  divisible  by  10. 

VI.  Any  number  is  divisible  by  3  if  the  sum  of  its  digits  is 
divisible  by  3. 

Thus,  582  is  divisible  by  3,  because  the  sum  of  its  digits, 
15,  is  divisible  by  3. 

VII.  Any  number  is  divisible  by  6  if  its  last  digit  is  0  or  an 
even  number^  and  the  sum  of  its  digits  is  divisible  by  3. 

This  follows  from  I  and  VI. 

VIII.  A7iy  number  is  divisible  by  9  if  the  sum  of  its  digits 
is  divisible  by  9. 


26  ARITHMETIC. 

Thus,  864  is  divisible  by  9,  because  the  sum  of  its  digits, 
18,  is  divisible  by  9. 

IX.  Any  number  is  divisible  by  11  if  the  sum  of  the  digits 
in  the  odd  places  is  equal  to  the  sum  of  the  digits  in  the  even 
places,  or  differs  from  it  by  a  number  divisible  by  11. 

Thus,  4785  is  divisible  by  11,  because  the  sum  of  the 
digits  in  the  first  and  third  places,  12,  is  equal  to  the  sum 
of  the  digits  in  the  second  and  fourth  places. 

Again,  39182  is  divisible  by  11,  because  the  sum  of  the 
digits  in  the  first,  third,  and  fifth  places,  6,  differs  from  the 
sum  of  the  digits  in  the  second  and  fourth  places,  17,  by  11 ; 
a  number  divisible  by  11. 

Note.     Principles  VI,  VIII,  and  IX  may  be  proved  as  follows : 

Proof  of  VI  and  VIII. 

Any  number  of  lO's  is  equal  to  the  same  number  of  9's,  plus  the 
same  number  of  units  ;  any  number  of  lOO's  is  equal  to  the  same  num- 
ber of  99's,  plus  the  same  number  of  units ;  etc. 

Thus,  783  is  equal  to  7  99's  plus  7  units,  8  9's  plus  8  units,  and  3 
units. 

But  the  sum  of  7  99's  and  8  9's  is  divisible  by  both  3  and  9. 

Hence,  783  is  divisible  by  3  or  9  if  the  sum  of  7  units,  8  units,  and 
3  units  is  divisible  by  3  or  9,  respectively  ;  that  is,  if  the  sum  of  its 
digits  is  divisible  by  3  or  9,  respectively. 

Similar  considerations  hold  with  respect  to  any  number. 

Proof  of  IX. 

Any  number  of  lO's  is  equal  to  the  same  number  of  ll's,  minus  the 
same  number  of  units ;  any  number  of  lOO's  is  equal  to  the  same 
number  of  99's  j9Z?/s  the  same  number  of  units ;  any  number  of  lOOO's 
is  equal  to*  the  same  number  of  lOOl.'s,  minus  the  same  number  of 
units ;  etc. 

Thus,  4829  is  equal  to  4  lOOl's  minus  4  units,  8  99's  plus  8  units, 
2  ll's  minus  2  units,  and  9  units. 

But  the  sum  of  4  lOOl's,  8  99's,  and  2  ll's,  is  divisible  by  11. 

Hence,  4829  is  divisible  by  11  if  the  sum  of  8  units  and  9  units, 
minus  the  sum  of  4  units  and  2  units,  is  divisible  by  11 ;  that  is,  if  the 
difference  between  the  sum  of  the  digits  in  the  odd  places  and  the  sum 
of  the  digits  in  the  even  places  is  divisible  by  11. 

Similar  considerations  hold  with  respect  to  any  number. 


FACTORING.  27 

57.   1.  Find  the  prime  factors  of  51480. 

2^)51480  51480  is  divisible  by  8,  or  23,  because  the  number 

5)6435         fornied  by  its  last  three  digits  is  divisible  by  8  (Art. 

3^)1287         ^^'  ^^^^' 

IvTZq  Dividing  51480  by  8,  the  quotient  is  6435. 

ll)14o  g435  |g  divisible  by  5,  because  its  last  digit  is  5. 

13  Dividing  6435  by  6,  the  quotient  is  1287. 

1287  is  divisible  by  9,  or  3^,  because  the  sum  of  its 
digits,  18,  is  divisible  by  9  (Art.  56,  VIII). 
Dividing  1287  by  9,  the  quotient  is  143. 

143  is  divisible  by  11,  because  the  sum  of  the  digits  in  the  first  and 
third  places  is  equal  to  the  digit  in  the  second  place  (Art.  56,  IX). 
Dividing  143  by  11,  the  quotient  is  13,  a  prime  number. 
Then,  51480  =  2^  x  3"^  x  5  x  11  x  13,  Ans. 

From  the  above  example,  we  derive  the  following 
RULE. 

Divide  the  number  by  any  one  of  its  factors;  then  divide 
the  quotient  by  any  one  of  its  factors;  and  so  on,  continuing 
the  process  until  the  quotient  is  a  prime  iiumber. 

The  several  divisors  and  the  last  quotient  are  the  factors 
required. 

Note  1.  In  determining  the  prime  factors  of  a  number,  divisors 
should  be  tried  in  the  following  order : 

2,  (22,  23);  5 ;  3,  (32);  11  ;  then,  7,  13,  17,  19,  23,  29,  etc. 

2.     Prove  that  373  is  a  prime  number. 

Since  the  last  digit  is  3,  neither  2  nor  5  is  a  factor. 

Since  the  sum  of  the  digits  is  13,  3  is  not  a  factor. 

Since  the  sum  of  the  digits  in  the  first  and  third  places  differs  from 
the  digit  in  the  second  place  by  1,  11  is  not  a  factor. 

Trying  in  order  the  prime  numbers  7,  13,  17,  and  19,  we  find  that 
neither  of  them  is  contained  in  373. 

There  is  no  need  of  trying  23,  nor  any  greater  prime  number ;  for 
if  one  factor  of  373  could  be  23,  the  other  would  have  to  be  23,  or 
some  greater  prime  number ;  for  it  has  already  been  proved  that  no 
prime  number  less  than  23  is  a  factor. 

But  23  X  23  =  529,  a  number  greater  than  373  ;  and  hence  23  can- 
not be  a  factor. 

Therefore  373  is  a  prime  number. 


28 


ARITHMETIC. 


Note  2.  Li  trying  prime  numbers  as  divisors,  the  process  need  not 
be  continued  after  the  square  of  tlie  prime  number  next  greater  than 
the  last  one  tried  exceeds  tlie  given  number. 

Tims,  in  Ex.  2,  no  prime  number  greater  than  19  need  be  tried, 
because  the  square  of  the  next  greater  prime,  23,  is  greater  than  378. 

Tlie  following  table  of  squares  of  prime  numbers  will  be  found  of 
use  : 


No. 

8q. 

No. 

Sq. 

No. 

Sq. 

No. 

Sq. 

No. 

Sq. 

13 

169 

29 

841 

43 

1849 

61 

3721 

79 

6241 

17 

289 

31 

961 

47 

2209 

67 

4489 

83 

6889 

19 

361 

37 

1369 

53 

2809 

71 

5041 

89 

7921 

23 

529 

41 

1681 

59 

3481 

73 

5329 

97 

9409 

58.   The  following  table  gives,  for  convenient  reference, 
a  list  of  the  prime  numbers  from  1  to  997,  inclusive : 

1  41  101  167  239  313   397  467  569   643  733  823  911 

2  43  103  173  241  317   401  479  571   647  739  827  919 

3  47  107  179  251  331  409  487  577  653  743  829  929 
5  53  109  181  257  337  419  491  587  659  751  839  937 
7  59  113  191  263  347   421  499  593   661  757  853  941 

11  61  127  193  269  349   431  503  599   673  761  857  947 

13  67  131  197  271  353   433  509  601   677  769  859  953 

17  71  137  199  277  359   439  521  607   683  773  863  967 

19  73  139  211  281  367   443  523  613   691  787  877  971 

23  79  149  223  283  373   449  541  617   701  797  881  977 

29  83  151  227  293  379   457  547  619   709  809  883  983 

31  89  157  229  307  383  461  557  631   719  811  887  991 

37  97  163  233  311  389  463  563  641   727  821  907  997 


EXAMPLES. 
59.   Find  the  prime  factors  of  the  following : 

1.  684.  5.   3003.  9.    1729.  13.  53625. 

2.  686.  6.   4459.  10.   8395.  14.  48204. 

3.  2520.  7.   1331.  11.   24108.  15.  32292. 

4.  4305.  8.   8085.     »     12.   19635.  16.  68364. 


FACTORING.  ^§ 

Prove  that  each  of  the  following  numbers  is  prime  : 

17.  1019.  19.   1367.  21.   1787.  23.   2203. 

18.  1193.  20.   1531.  22.   2081.  24.   2999. 

60.  Verification  of  Addition,  Subtraction,  Multiplication, 
and  Division  by  Casting  out  Nines. 

The  excess  of  any  number  above  the  next  less  multiple 
of  9  is  called  its  Excess  of  Nines. 

Thus,  since  3527  is  equal  to  391  x  9,  plus  8,  its  excess  of 
nines  is  8. 

It  was  shown  in  the  proof  of  YIII  (Art.  56,  Note)  that 
any  number  is  equal  to  a  multiple  of  9,  plus  the  sum  of  its 
digits. 

Thus,  3527  is  equal  to  a  multiple  of  9,  plus  17 ;  or  since 
17  =  9  +  8,  it  is  equal  to  a  multiple  of  9,  plus  8. 

It  follows  from  the  above  that  the  excess  of  nines  of  any 
number  may  be  found  by  subtracting  from  the  sum  of  its 
digits  the  next  less  multiple  of  9. 

Thus,  since  the  sum  of  the  digits  of  the  number  4619  is 
20,  its  excess  of  nines  is  20  —  18,  or  2. 

Addition. 

3527...    8 
4619...    2 


1...8146      10...  1 

The  excess  of  nines  of  3527  is  8,  and  of  4619  is  2. 

Then  since  3527  is  equal  to  a  multiple  of  9,  plus  8,  and 
4619  to  a  multiple  of  9,  plus  2,  their  sum,  is  equal  to  a  mul- 
tiple of  9,  plus  8,  plus  2 ;  or  since  10  =  9  +  1,  it  is  equal  to 
a  multiple  of  9,  plus  1. 

But  the  excess  of  nines  of  8146  is  1;  that  is,  8146  is 
equal  to  a  multiple  of  9,  plus  1. 

This  agrees  with  the  statement  made  in  the  preceding 
paragraph  with  respect  to  the  sum  of  3527  and  4619. 


80  ARITHMETIC. 

Then,  to  verify  the  result  of  addition,  we  place  to  the 
right  of  each  of  the  numbers  to  be  added  its  excess  of  nines. 

Adding  these  excesses,  we  place  to  the  right  of  their  sum 
its  excess  of  nines. 

If  this  equals  the  excess  of  nines  of  the  result,  the  work 
may  be  considered  to  be  correct. 

Subtraction. 

6782...  5  2235...  3...  12 

2946. ..3  1167  ...    6 


2...  3836      2  6...  1068  6 

To  the  right  of  the  minuend  and  subtrahend  we  place 
their  excess  of  nines. 

We  then  subtract  the  excess  of  the  subtrahend  from  that 
of  the  minuend,  increasing  the  latter  by  9  if  it  is  less  than 
the  excess  of  the  subtrahend. 

If  the  remainder  equals  the  excess  of  nines  of  the  result, 
the  work  may  be  considered  to  be  correct, 


Multiplication. 


498...    3 
376...    7 

2988   21  ...3 
3486 
1494 


3...  187248 

Since  498  is  equal  to  a  multiple  of  9,  plus  3,  and  376  to  a 
multiple  of  9,  plus  7,  their  product  is  equal  to  the  result 
obtained  by  multiplying  a  multiple  of  9  plus  3,  first  by  a 
multiple  of  9,  and  afterwards  by  7,  and  adding  the  results. 

It  is  evident  from  this  that  the  product  is  equal  to  a 
multiple  of  9,  plus  3  times  7 ;  or  since  21  =  18  +  3,  it  is 
equal  to  a  multiple  of  9,  plus  3. 

To  the  right  of  each  of  the  numbers  to  be  multiplied  we 
place  its  excess  of  nines. 


FACTORING.  31 

Multiplying  the  excess  of  the  multiplicand  by  that  of  the 
multiplier,  we  place  to  the  right  of  the  product  its  excess 
of  nines. 

If  this  equals  the  excess  of  nines  of  the  result,  the  work 
may  be  considered  to  be  correct. 

Division. 


(768.. 

.3 

3 

329)252784.. 

.1 

5 

2303 

15 

2248 

4 

1974 

19 

2744 

2632 

112  ...4 

Since  the  dividend  is  equal  to  the  product  of  the  divisor 
and  quotient,  plus  the  remainder,  we  proceed  as  follows : 

Multiply  the  excess  of  nines  of  the  quotient  by  that  of 
the  divisor. 

Add  to  the  product  the  excess  of  nines  of  the  remainder, 
and  place  to  the  right  of  the  sum  its  excess  of  nines. 

If  this  equals  the  excess  of  nines  of  the  dividend,  the 
work  may  be  considered  to  be  correct. 

Note.  The  above  methods  are  not  always  .to  be  depended  upon  as 
tests  of  the  accuracy  of  operations. 

Suppose,  for  instance,  that,  in  the  illustrative  example  under  Addi- 
tion, we  had  taken  the  sum  of  2  and  1  as  4,  and  the  sum  of  5  and  6 
as  10: 

3527 
4619 


8056 


The  excess  of  nines  in  the  sum  will  still  be  1,  and  yet  the  work  is 
not  performed  correctly. 

A  balance  of  errors  like  this  is,  however,  unlikely  to  occuj. 


32  ARITHMETIC. 

61.  Casting  out  Elevens. 

The  excess  of  any  number  above  the  next  less  multiple 
of  11  is  called  its  Excess  of  Elevens. 

Thus,  since  3527  is  equal  to  320  x  11,  plus  1,  its  excess 
of  elevens  is  7. 

It  was  shown  in  the  proof  of  IX  (Art.  5Q>,  Note)  that  any 
number  is  equal  to  a  multiple  of  11,  plus  the  sum  of  the 
digits  in  the  odd  places,  minus  the  sum  of  the  digits  in  the 
even  places. 

Thus,  3527  is  equal  to  a  multiple  of  11,  plus  12,  minus  5 ; 
or,  to  a  multiple  of  11,  plus  7. 

It  follows  from  the  above  that  the  excess  of  elevens  of 
any  number  may  be  found  by  subtracting  the  sum  of  the 
digits  in  the  even  places  from  the  sum  of  the  digits  in  the  odd 
places^  the  latter  being  increased  by  11,  or  a  multiple  of  11,  if 
necessary. 

Thus,  for  the  number  9484,  the  sum  of  the  digits  in  the 
odd  places  is  8,  and  the  sum  of  the  digits  in  the  even  places 
is  17. 

Increasing  the  former  by  11,  we  have  19. 

Then  the  excess  of  elevens  is  19  — 17,  or  2. 

The  methods  for  verifying  Addition,  Subtraction,  Multi- 
plication, and  Division  by  casting  out  elevens  are  precisely 
similar  in  theory  and  practice  to  the  methods  by  casting  out 
nines. 


GREATEST   COMMON  DIVISOR.  33 


VII.    GREATEST  COMMON  DIVISOR. 

62.  A  Common  Divisor,  or  Common  Factor,  of  two  or  more 
whole  numbers  is  a  number  that  will  divide  each  of  them 
without  a  remainder. 

Thus,  3  is  a  common  divisor  of  18,  24,  and  30. 

63.  The  Greatest  Common  Divisor  (G.  C.  D.)  of  two  or 

more  whole  numbers  is  the  greatest  number  that  will  divide 
each  of  them  without  a  remainder. 

Thus,  6  is  the  greatest  common  divisor  of  18,  24,  and  30. 

64.  Two  numbers  are  said  to  be  prime  to  each  other  when 
they  have  no  common  divisor  except  1. 

Thus,  8  and  9  are  prime  to  each  other. 

65.  In  determining  the  G.  C.  D.  of  numbers,  we  may  dis- 
tinguish two  cases : 

66.  Case  I.  When  the  numbers  can  he  readily  factored  as 
in  Art-  57. 

1.   Find  the  G.  C.  D.  of  144,  264,  and  540. 

144  =  2^  X  3^  Factoring  each  number  by  the  method 

264  =  2^  X  3  X  11  of  Art.  57,  it  is  evident  that  the  greatest 

540  =  2^  X  3^  X  5  number  that  will  exactly  divide  144,  264, 

G.  C.  D.  =  2^  X  3  and  540,  is  22  x  3. 

=  12,  Ans.  Hence,  the  required  G.  C.  D.  is  12. 

From  the  above  example  we  derive  the  following 

RULE. 

Factor  each  of  the  numbers. 

Take  every  prime  number  which  is  a  common  divisor  of  all 
the  given  numbers,  the  least  number  of  times  that  it  occurs  in 
any  one  of  the  numbers. 

The  product  of  these  numbers  will  be  the  G.  C.  D.  required. 


34  ARITHMETIC. 

Note.  If  any  prime  number  is  a  common  divisor  of  all  the  given 
numbers,  its  exponent  in  the  G.  C.  D.  will  be  the  lowest  exponent  with 
which  it  occurs  in  any  one  of  the  numbers. 

Thus,  m  Ex.  1 ,  we  have  in  the  given  numbers  2*,  2^,  and  2^  respec- 
tively, and  m  the  G.  C.  D.,  22. 

If  one  of  the  numbers  is  exactly  contained  in  another,  the 
latter  need  not  be  considered  in  the  operation  of  finding 
the  G.  C.  D. ;  for  since  every  factor  of  the  first  number  is 
also  a  factor  of  the  second,  the  result  is  not  affected  by 
omitting  the  second  number  from  the  process. 

Thus,  in  finding  the  G.  C.  D.  of  28,  49,  140,  and  196,  it 
would  be  sufficient  to  find  the  G.  C.  D.  of  28  and  49. 

EXAMPLES. 
Find  the  G.  C.  D.  of: 

2.  165  and  210.  12.  104,  182,  and  351. 

3.  288  and  648.  13.  180,  264,  and  378. 

4.  306  and  476.  14.  16,  52,  160,  224,  and  260. 

5.  36,  144,  and  234.  15.  320,  640,  and  1008. 

6.  128,  192,  and  384.  16.  390,  910,  and  1365. 

7.  675  and  1125.  17.  360,  750,  and  2700. 

8.  105,  385,  and  455.  18.  432,  1944,  and  2592. 

9.  96,  108,  132,  and  156.  19.  525,  3375,  and  7425. 

10.  240,  336,  and  480.  20.    1540,  5005,  and  6545. 

11.  81,  117,  126,  and  135.    21.   6804,  7056,  and  8232. 

22.  A  farmer  has  three  pieces  of  timber  whose  lengths 
are  63,  84,  and  105  feet,  respectively.  What  is  the  length 
of  the  longest  logs,  all  of  the  same  length,  that  can  be  cut 
from  them  ? 

23.  Two  schools,  containing  480  and  672  pupils,  respec- 
tively, are  divided  into  classes,  each  containing  the  same 
number  of  pupils.  What  is  the  greatest  number  of  pupils 
that  each  class  can  contain,  and  how  many  classes  of  this 
size  are  there  in  each  school  ? 


GREATEST  COMMON  DIVISOR.  35 

24.  Three  rooms  are  168,  196,  and  224  inches  wide,  re- 
spectively. What  is  the  width  of  the  widest  carpeting  that 
is  contained  exactly  in  each  room  ? 

25.  How  many  quarts  are  there  in  the  largest  receptacle 
that  will  exactly  measure  the  contents  of  three  jars,  holding 
216,  288,  and  312  quarts,  respectively  ? 

26.  I  have  three  fields  containing  392,  504,  and  616 
square  rods,  respectively.  Find  the  size  of  the  largest 
house-lots,  all  of  the  same  size,  into  which  the  fields  can  be 
divided. 

27.  The  sides  of  a  field  are  110,  154,  198,  and  264  feet, 
respectively.  What  is  the  length  of  the  longest  fence-rail 
that  is  contained  exactly  in  each  side  ? 

67.  Case  II.  Whe^i  the  numbers  cannot  be  readily  fac- 
tored as  in  Art.  57. 

1.   Find  the  G.  C.  D.  of  221  and  493. 

Dividing  the  greater  number  by  the  less,  we  have 

221)493(2,  Quotient. 
442 

51,  Kemainder. 

Now  whatever  factors  occur  in  221  must  also  occur  in 
twice  221,  or  442. 

Hence,  any  factors  which  are  common  to  221  and  493 
must  occur  in  the  result  obtained  by  subtracting  442  from 
493 ;  that  is,  they  must  occur  in  51. 

Again,  any  factors  which  are  common  to  221  and  51 
must  occur  in  the  result  obtained  by  adding  twice  221  to 
51 ;  that  is,  they  must  occur  in  493. 

Then,  since  every  factor  common  to  221  and  493  occurs 
in  51,  and  every  factor  common  to  221  and  51  occurs  in 
493,  it  follows  that  221,  493,  and  51  have  the  same  common 
factors. 


36  ARITHMETIC. 

Hence,  the  G.  C.  D.  of  221  and  493  must  be  the  same  as 
the  G.  C.  D.  of  221  and  51. 

That  isj  the  G.  C.  D.  of  any  tiuo  numbers  is  the  same  as  the 
O.  C.  D.  of  the  less  number,  and  the  remainder  obtained  by 
dividing  the  greater  number  by  the  less. 

Dividing  the  divisor,  221,  by  the  remainder,  51,  we  have 
51)221(4 
204 
17,  Eemainder. 
Then,  by  the  principle  just  stated,  the  G.  C.  D.  of  221  and 
51  is  the  same  as  the  G.  C.  D.  of  51  and  17. 

Dividing  the  divisor,  51,  by  the  remainder,  17,  we  have 

17)51(3 

51 

That  is,  17  is  the  G.  C.  D.  of  51  and  17. 
Then,  17  is  also  the  G.  C.  D.  of  221  and  51,  and  is  conse- 
quently the  G.  C.  D.  of  493  and  221. 
From  the  above  example,  we  derive  the  following 

RULE. 
Divide  the  greater  number  by  the  less. 

If  there  be  a  remainder,  divide  the  divisop  by  it;  and  con- 
tinue thus  to  make  the  remainder  the  divisor,  and  the  preceding 
divisor  the  dividend,  imtil  there  is  no  remainder. 
The  last  divisor  is  the  O.  C.  D.  required. 
2.   Find  the  G.  C.  D.  of  377  and  667. 
377)667(1 
377 

290)377(1 
290 
87)290(3 
261 
29)87(3 
87 

Then,  29  is  the  G.  C.  D.  required,  Ans. 


GREATEST   COMMON   DIVISOR.  37 

EXAMPLES. 
Find  the  G.  C.  D.  of: 

3.  559  and  817.  10.  3703  and  6923. 

4.  391  and  598.  11.  1591  and  2183. 

5.  589  and  899.  12.  5605  and  6785. 

6.  703  and  893.  13.  6059  and  7446. 

7.  533  and  1271.  14.  5312  and  10043. 

8.  731  and  1247.  15.  2291  and  3713. 

9.  3658  and  4602.  16.  10057  and  11659. 

68.  The  G.  C.  D.  of  three  numbers  which  cannot  be  readily- 
factored  by  the  method  of  Art.  57,  may  be  found  as  follows : 

Let  A,  B,  and  C  represent  the  numbers. 

Let  G  represent  the  G.  C.  D.  of  A  and  B ;  then  every 
common  factor  of  G  and  C  is  also  a  common  factor  of  A,  B, 
and  C. 

But  every  common  factor  of  A  and  B  exactly  divides  G. 

Whence,  every  common  divisor  of  A,  B,  and  C  is  also  a 
common  divisor  of  G  and  C. 

Therefore,  the  greatest  common  divisor  of  A,  B,  and  G  is 
the  same  as  the  greatest  common  divisor  of  G  and  C. 

Hence,  to  find  the  G.  C.  D.  of  three  numbers,  find  the 
G.  C.  D.  oj  two  of  them,  and  then  of  this  result  and  the  third 
number. 

We  proceed  in  a  similar  manner  to  find  the  G.  C.  D.  of 
four  or  more  numbers. 

1.   Find  the  G.  C.  D.  of  741,  1653,  and  7163. 

We  first  find  the  G.  C.  D.  of  741  and  1653,  which  is  57. 
We  then  find  the  G.  C.  D.  of  57  and  7163,  which  is  19,  Ans. 

EXAMPLES.  •- 

Find  the  G.  C.  D.  of: 

2.  663,  741,  and  4199.      4.  969,  1653,  and  9367. 

3.  442,  782,  and  5083.      5.  5083,  5681,  and  7429. 


88  ARITHMETIC. 


VIII.    LEAST  COMMON  MULTIPLE. 

69.  A  Cominon  Multiple  of  two  or  more  whole  numbers 
is  a  member  that  will  exactly  contain  each  of  them. 

Thus,  72  is  a  common  multiple  of  6,  9,  and  12. 

70.  The  Least  Common  Multiple    (L.  C.  M.)    of  two   or 

more  whole  numbers  is  the  smallest  number  that  will  exactly 
contain  each  of  them. 

Thus,  36  is  the  least  common  multiple  of  6,  9,  and  12. 

71.  In  determining  the  L.  C.  M.  of  numbers,  we  may  dis- 
tinguish two  cases : 

72.  Case  I.     When  the  numbers  can  be  readily  factored. 

Example.     Find  the  L.  C.  M.  of  40,  84,  and  144. 

40  _  93           y  5  Factoring  each  of  the  numbers  by 

Q/l  _  92      Q               7  *^^  method  of  Art.  57,  it  is  evident 

»4  =  J   X  o           X  7  ^-^^^  ^-^^  smallest  number  that  will 

^^^  —  2^^X^3^ exactly   contain  40,  84,  and  144,  is 

L.  C.  M.  =  2^  X  32  X  5  X  7      24  X  32  X  5  X  7. 

w^.^      .  Hence,   the  required  L.  C.  M.   is 

=  5Q^,Ans.  ^^^ 

From  the  above  example,  we  derive  the  following 

RULE. 

Factor  each  of  the  numbers. 

Take  every  prime  number,  which  is  a  factor  of  any  one  of 
the  given  numbers.,  the  greatest  number  of  times  that  it  occurs 
in  any  one  of  the  numbers. 

The  product  of  these  numbers  will  be  the  L.  C.  M.  required. 

Note.  If  aj;iy  prime  number  is  a  factor  of  any  one  of  the  given 
numbers,  its  exponent  in  the  L.  C.  M.  will  be  the  greatest  exponent 
with  which  it  occurs  in  any  one  of  the  numbers. 

Thus,  in  the  above  example,  we  have  in  the  given  numbers  2^,  22, 
and  2*,  respectively,  and  in  the  L.  C.  M.,  2*. 


LEAST   COMMON  MULTIPLE.  39 

73.   Second  Method. 

The  following  rule  will  be  found  preferable  to  that  of 
Art.  72  in  the  solution  of  examples  : 

Arrange  the  numbers  in  a  horizontal  line. 

If  two  or  more  of  the  numbers  have  a  common  prime  factor, 
divide  them  by  it,  and  ivrite  the  quotients,  together  with  the 
undivided  numbers,  in  the  next  line. 

Continue  in  this  way  until  a  line  is  obtained  m  which  the 
numbers  have  no  common  factor. 

The  prodiLCt  of  the  divisors  and  the  numbers  in  the  last  line 
will  be  the  L.  C.  M.  required. 

1.   Find  the  L.  C.  M.  of  24,  60,  and  105. 

2)24        60        105  Dividing  24  and  60  by  the  com- 

mon prime  factor  2,  the  second  line 
becomes  12,  30,  105. 

Dividing  12  and  30  by  the  com- 
mon prime  factor  2,  the  third  line 
2  17  becomes  6,  15,  105. 

L.  C.  M.  =  2x2x3x5x2x7       Dividing  6,  15,  and  105  by  the 
=  840   Ans.  common  prime  factor  3,  the  fourth 

line  becomes  2,  5,  35. 
Dividing  5  and  35  by  the  common  prime  factor  5,  the  fifth  line 
becomes  2,  1,  7. 

Since  the  numbers  2  and  7  have  no  common  prime  factor,  the 
required  L.  C.  M.  is  the  product  of  the  divisors,  2,  2,  3,  5,  and  the 
numbers  in  the  last  line,  2,  7  ;  the  result  is  840. 

It  is  evident  that,  *in  the  above  process,  every  prime  num- 
ber which  is  a  factor  of  any  one  of  the  given  numbers,  is 
taken  the  greatest  number  of  times  that  it  occurs  in  any  one 
of  the  numbers. 

Hence,  the  result  is  the  L.  C.  M.  of  the  given  numbers. 

If  one  of  the  given  numbers  exactly  divides  another,  the 
former  need  not  be  considered  in  the  operation  of  finding 
the  L.  C.  M. ;  for  since  every  factor  of  the  first  number  is 
also  a  factor  of  the  second,  the  result  is  not  affected  by 
omitting  the  first  number  from  the  process. 


012 

30 

105 

3)6 

15 

105 

5)2 

5 

35 

40  ARITHMETIC. 

Tims,  in  finding  the  L.  C.  M.  of  15,  26,  78,  and  90,  it  would 
be  sufficient  to  find  the  L.  C.  M.  of  78  and  90. 

Note.  If  two  numbers  are  prime  to  each  other  (Art.  64),  their 
product  is  their  L.  C.  M. 

EXAMPLES. 
Find  the  L.  CM.  of: 

2.  4,  6,  9,  and  10.  14.  28,  49,  147,  and  196. 

3.  28  and  63.  15.  24,  42,  72,  84,  and  112. 

4.  24,  112,  and  160.  16.  15,  35,  and  77. 

5.  108  and  144.  17.  36,  104,  and  351. 

6.  110  and  165.  18.  115,  138,  230,  and  345. 

7.  231  and  770.  19.  33,  44,  55,  and  132. 

8.  18,  38,  54,  and  57.  20.  288,  324,  432,  and  648. 

9.  20,  75,  180,  and  300.  21.  189,  243,  and  405. 

10.  176  and  264.  22.  98,  126,  140,  and  168. 

11.  32,  88,  and  121.  23.  52,  81,  117,  and  120. 

12.  87,  116,  and  192.  24.  119,  136,  252,  and  280. 

13.  144,  216,  and  324.  25.  315,  1350,  and  1500. 

26.  How  many  quarts  are  there  in  the  smallest  vessel 
whose  contents  can  be  exactly  measured  by  measures  con- 
taining 10,  15,  and  18  quarts,  respectively  ? 

27.  Two  horse-cars  make  round  trips  in  48  and  60  min- 
utes, respectively.  If  they  set  out  at  the  same  time,  after 
how  many  minutes  will  they  meet  again  at  the  starting- 
point  ? 

28.  What  is  the  smallest  sum  of  money  with  which  I  can 
pjirchase  cows  at  $  45  each,  oxen  at  $  54  each,  or  horses  at 
f  72  each  ? 

29.  Three  men.  A,  B,  and  C,  can  walk  around  a  race- 
course in  9,  12,  and  14  minutes,  respectively.  If  they  all 
set  out  at  the  same  time,  after  how  many  minutes  will  they 
all  meet  at  the  starting-point,  and  how  many  times  will 
each  have  been  around  the  course  ? 


LEAST  COMMON   MULTIPLE.  41 

74.  Case  II.    When  the  numbers  cannot  be  readily  factored. 

1.  Find  the  L.  C.  M.  of  221  and  247. 

If  we  divide  221  by  the  greatest  common  divisor  of  221 
and  247,  the  quotient  will  be  the  product  of  those  factors 
of  221  which  are  not  found  in  247. 

Then,  if  we  multiply  this  quotient  by  247,  the  product 
will  be  divisible  by  both  221  and  247 ;  and  it  is  evidently 
the  smallest  number  that  is  divisible  by  both  of  them. 

That  is,  the  product  is  the  L.  C.  M.  of  221  and  247. 

221)247(1  13)221(17  247 

221  13_  _17 

26)221(8  91  1729 

208  91  247^ 

G.  C.  D.  =  13)26(2  L.  C.  M.=4199,  Ans. 

26 

•     We  first  find  the  G.  C.  D  of  221  and  247  by  the  rule  of  Art.  67  ;  the 
result  is  13. 

Dividing  221  by  13,  the  quotient  is  17. 

Multiplying  247  by  17,  the  product  is  4199. 

Then,  the  required  L.  C.  M.  is  4199. 

From  the  above  example,  we  derive  the  following 

RULE. 
Find  the  O.  C.  D.  of  the  given  numbers. 
Divide  one  of  the  numbers  by  their  G.  C.  />.,  and  multiply 
the  quotient  by  the  other  number. 

EXAMPLES. 
Find  the  L.  C.  M.  of : 

2.  289  and  323.  8.  361  and  437. 

3.  629  and  703.  9.  391  and  493. 

4.  551  and  589.  10.  403  and  961. 

5.  667  and  713.  11.  533  and  1189. 

6.  841  and  899.  12.  1403  and  1817. 

7.  299  and  529.  13.  6649  and  7957. 


42  ARITHMETIC. 

75.  The  L.  C.  M.  of  three  numbers  which  cannot  be  readily 
factored,  may  be  found  as  follows : 

Let  Ay  B,  and  C  represent  the  numbers. 

Let  M  represent  the  L.  C.  M.  of  A  and  B ;  then,  every 
common  multiple  of  M  and  C  is  also  a  common  multiple  of 
A,  B,  and  C. 

But  every  common  multiple  of  A  and  B  exactly  con- 
tains M. 

Whence,  every  common  multiple  of  A,  B,  and  C  is  also  a 
common  multiple  of  M  and  C. 

Therefore,  the  least  common  multiple  of  A,  B,  and  C  is  the 
same  as  the  least  common  multiple  of  M  and  C. 

Hence,  to  find  the  L.  C.  M.  of  three  numbers,  fiyid  the  L.  C.  M. 
of  two  of  them,  and  then  of  this  result  and  the  third  number. 

We  proceed  in  a  similar  manner  to  find  the  L.  C.  M.  of 
four  or  more  numbers. 

1.  Find  the  L.  C.  M.  of  713,  1081,  and  1395. 

We  first  find  the  L.  C.  M.  of  713  and  1081,  which  is  33511. 

We  then  find  the  L.  C.  M.  of  1395  and  33511,  which  is  1607995,  Ans. 

EXAMPLES.  ' 

Find  the  L.  C.  M.  of: 

2.  1271,  1674,  and  1968.     3.  1505, 1591,  and  2109. 


FRACTIONS.  48 


IX.    FRACTIONS. 

76.  If  unity  is  divided  into  4  equal  parts,  and  3  parts  are 
taken,  the  result  is  expressed  by  j ;  read  "  three-fourths.^^ 

If  unity  is  divided  into  any  number  of  equal  parts,  and 
any  number  of  parts  are  taken,  the  result  is  called  a 
Fraction. 

77.  The  Denominator  of  a  fraction  is  the  number  which 
shows  into  how  many  equal  parts  unity  is  divided,  and  the 
Numerator  is  the  number  which  shows  how  many  parts  are 
taken. 

Thus,  in  the  fraction  f ,  the  denominator  is  4,  and  the 
numerator  is  3. 

The  numerator  and  denominator  are  called  the  Terms  of 
the  fraction. 

78.  A  fraction  is  usually  expressed  by  writing  the  numer- 
ator above,  and  the  denominator  below,  a  horizontal  line; 
and  when  thus  expressed,  it  is  called  a  Common  Fraction. 

79.  A  Mixed  Number  is  the  sum  of  a  whole  number  and 
a  fraction. 

Thus,  5  +  1,  or,  as  it  is  usually  written,  5f ,  is  a  mixed 
number. 

80.  Let  each  of  the  lines  AB,  BC,  and  CD,  in  the  follow- 
ing figure,  represent  one  unit;  then  AD  will  represent  3 
units. 

A      E      F      G      B  G  D 

I         I        I          [         I         I        I         I         I         1         I         I         I 

Let  AB  be  divided  into  4  equal  parts ;  AE,  EF,  FG,  and 
OB\  then  the  fraction  f  will  be  represented  by  AG. 

Now  it  is  evident  that,  if  AD  be  divided  into  4  equal 
parts,  one  of  these  parts  will  be  AG. 


44  ARITHMETIC. 

Hence,  the  fraction  J  represents  the  result  obtained  by- 
dividing  3  units  by  4. 

And  in  general,  any  fraction  is  an  expression  of  division; 
the  numerator  answering  to  the  dividend,  and  the  denomi- 
nator to  the  divisor. 

81.  It  follows  from  Art.  80  that  an  integer  may  be  ex- 
pressed in  a  fractional  form  by  writing  1  for  a  denominator. 

Thus,  3  is  the  same  as  f . 

82.  A  Proper  Fraction  is  one  whose  numerator  is  less 
than  its  denominator ;  as  J. 

An  Improper  Fraction  is  one  whose  numerator  is  equal  to 
or  greater  than  its  denominator ;  as  |-,  or  ^. 

REDUCTION  OF  FRACTIONS. 

83.  To  Reduce  an  Improper  Fraction  to  a  Whole  or  Mixed 
Number. 

1.  Eeduce  ^-  to  a  whole  number. 

Since  a  fraction  is  an  expression  of  division  (Art.  80), 
^  =  54-9  =  6,^718. 

2.  Reduce  ^^  to  a  mixed  number. 

Since  290  is  equal  to  the  sum  of  276  and  14,  we  have 

W  =  W  +  M  =  12  +  i|,orl2i|,  ^ns. 

It  is  customary  to  perform  the  work  as  follows : 

23)290(12^,  Ans. 
23 
60 
46 
14,  Remainder. 

From  the  above  examples,  we  derive  the  following 

RULE. 
Divide  the  numerator  by  the  denominator. 
If  there  is  a  remainder,  write  it  over  the  divisor,  and  add 
the  fraction  thus  formed  to  the  quotient. 


FRACTIONS.  45 


EXAMPLES. 

Reduce  each  of  the  following  to  a  whole  or  mixed  number : 

3.  M-  7.  ifF-  11.  ^iF-  15-  ^W^. 

4.  ^.  8.  m^.  12.  ^ffi.  16.  ifll-^- 

6.    ^^.  10.    ^A.  14.    i^||5.  18.    1^403. 

84.  To  Reduce  a  Whole  Number  to  a  Fraction  having  a 
given  Denominator. 

1.  Reduce  5  to  sevenths. 

Since  1  is  equal  to  7  sevenths,  5  is  equal  to  5  times  7  sevenths,  or 
35  sevenths  ;  whence,  5  =  Y-,  Ans. 

From  the  above  example,  we  derive  the  following 

RULE. 

To  reduce  a  whole  number  to  a  fraction  having  a  given 
denominator,  multiply  the  whole  number  by  the  denominator, 
and  write  the  result  as  the  numerator  of  the  required  fraction. 

EXAMPLES. 

2.  Reduce  6  to  8ths.  6.  Reduce  22  to  18ths. 

3.  Reduce  13  to  6ths.  7.  Reduce  19  to  15ths. 

4.  Reduce  11  to  9ths.  8.  Reduce  31  to  24ths. 

5.  Reduce  16  to  12ths.  9.  Reduce  48  to  37ths. 

85.  To  Reduce  a  Mixed  Number  to  an  Improper  Fraction. 

1.  Reduce  9|-  to  an  improper  fraction. 

Since  9  is  equal  to  72  eighths,  9|  is  equal  to  the  sum  of  72  eighths 
and  7  eighths,  which  is  79  eighths  ;  whence,  9|  =  ^^-,  Ans. 

From  the  above  example,  we  derive  the  following 

RULE. 
Multiply  the  whole    number   by   the   denominator  of   the 
fraction;  add  to  the  product  the  numerator  of  the  fraction, 
and  write  the  result  over  the  given  denominator. 


46  ARITHMETIC. 

EXAMPLES. 
Reduce  each  of  the  following  to  an  improper  fraction 


2.  81^. 

6.  7|f. 

10.  54i|. 

14.  74U. 

3.  lOi 

7.  9«. 

U.  29fi. 

15.  96|f. 

4.  IIJ,. 

8.  40tV. 

12.  58f|. 

16.  127fJ. 

5.  13,^. 

9.  79ff . 

13.  37^. 

17.  156J|. 

86.  To  Reduce  a  Fraction  to  its  Lowest  Terms. 

A  fraction  is  said  to  be  in  its  Lowest  Terms  when  its 
numerator  and  denominator  have  no  common  factor. 

87.  Let  the  line  AO,  in  the  following  figure,  represent 
one  unit ;  and  let  it  be  divided  into  6  equal  parts,  AB,  BC, 
CD,  DE,  EF,  and  FG. 

A  B  C  T>  :E  F  Q 

b— ^— M^M^W^— lid  I  I 

tori  1 

Then  the  fraction  ^  will  be  represented  by  AE. 

But  since  the  divisions  AC,  CE,  and  EG  are  all  equal, 
the  line  AE  also  represents  the  fraction  -|. 

Hence,  the  fraction  f  is  equal  to  |. 

Now  the  fraction  f  may  be  obtained  from  ^  by  dividing 
both  numerator  and  denominator  by  2. 

Hence,  if  both  numerator  and  denominator  of  ^  be  divided 
by  2,  the  value  of  the  fraction  is  not  changed. 

And  in  general,  if  both  numerator  and  denominator  of  any 
fraction  he  divided  by  the  same  number,  the  value  of  the  frac- 
tion is  not  changed. 

88.  In  reducing  fractions  to  their  lowest  terms,  we  may 
distinguish  two  cases : 

89.  Case  I.  When  the  numerator  and  denominator  can  be 
readily  factored. 


FRACTIONS.  47 

Since  both  numerator  and  denominator  can  be  divided  by 
the  same  number  without  changing  the  value  of  the  fraction 
(Art.  87),  we  have  the  following 

RULE. 

Divide  both  numerator  and  denominator  by  any  common 
factor. 

The  greater  the  common  divisor  used,  the  more  rapid 
will  be  the  process. 

1.  Eeduce  ^-|f  to  its  lowest  terms. 

i9^_   9  9   _  33  _  11     J^o  Dividing  both  terms  of  ^||  by  2, 

1Z2-T2Q-T2-  TT'  ^^«-     the  result  is  j%\. 

Dividing  both  terms  of  y^g  by  3,  the  result  is  ||. 

Dividing  both  terms  of  f  |  by  3,  the  result  is  }|. 

If  all  the  factors  of  the  numerator  be  removed  by  division, 
1  remains  to  form  a  numerator. 

If  all  the  factors  of  the  denominator  be  removed,  the 
result  is  a  whole  number,  this  being  a  case  of  exact  division. 

EXAMPLES. 
Reduce  each  of  the  following  to  its  lowest  terms : 

2-  AV       6.  in.        10.  ^^.       14.  mi- 

S        22  7        288  11         6  50 


4-  W-  8.  m-  12- 

6-  m-         9-  W/-  13- 


1936 
2662- 


Tl 


15. 

2700 

1125' 

16. 

2592 
5832- 

17. 

6615 

iJ646- 

90.  Cancellation. 

Cancellation  is  the  process  of  dividing  both  numerator 
and  denominator  by  striking  out  their  common  factors. 

It  is  useful  in  cases  when  either  the  numerator  or  de- 
nominator is  expressed  in  the  form  of  a  product. 


48  ARITHMETIC. 

1.  Reduce  ^l^]^^  ^^  to  its  lowest  terms. 
20  X  14  X  36 

Cancelling  7  from  21  and  14, 

^  S  we  write  3  above  21,  and  2  be- 

nxX^xl^^     3     _3    ^^^      10W14. 

^flXl^X^^      4x2      8'  '         Cancelling  5  from  15  and  20, 

4         2^  we  write  3  above  15,   and  4 

below  20. 

Cancelling  12  from  12  and  36,  we  write  3  below  36. 

We  then  cancel  the  3  above  21  with  the  3  below  36. 

3  3 

The  result  is  ,  or  — 

4x2         8 


EXAMPLES. 

Reduce  each  of  the  following  to  its  lowest  terms  : 
»     49  X  88  y     8  X  12  X  14  X  15 


55  X  112 

46x63 
7  X  23  X  30' 


8. 


10 

Xl6 

X  18  X  21 

34 

X38 

x39 

57 

X85 

x91 

27 

X77 

Xl05 

135  X 

165 

21 

x26 

x52 

39 

x56 

Xll7 

54 

x84 

x270 

.  20    X    108  g 

■    18x28x36 

K    16  X  95  X  96  .Q 

114x128 

g    21  X  39  X  55*  .. 

20  X  26  X  33*  *    50  x  162  x  196 

91.  Case  II.  When  the  numerator  and  denominator  can- 
not be  readily  factored. 

Since  the  G.  C.  D.  of  the  numerator  and  denominator  is 
the  greatest  number  that  will  exactly  divide  each  of  them, 
we  have  the  following 

RULE. 

Divide  both  numerator  and  denominator  by  their  greatest 
common  divisor. 


FRACTIONS.  49 

1.   Reduce  ||-|  to  its  lowest  terms. 

247)323(1 
247 

76)247(3 

228  ^6  fi^d'  ^y  t^6  rule  of  Art.  67,  that 

"^rr^^  Dividing  247  by  19,  the  quotient  is 

—  13. 

19)247(13         19)323(17     ^^^^^^^^^  ^23  by  19,  the  quotient  is 


19_  19^ 

57  133 

57  133 

1^,  Ans. 


Then  the  required  result  is  ^f . 


EXAMPLES. 

Eeduce  each  of  the  following  to  its  lowest  terms : 
2-  m-        4-  m-        6.  iflf .        8.  Vi^.        10.  Mff. 
5.  Iff        7.  Itti-        9-  liH-        11-  Mfi- 


92.  To  Reduce  Fractions  to  their  Least  Common  Denomi- 
nator. 

Fractions  are  said  to  have  a  Common  Denominator  when 
they  all  have  the  same  denominator. 

To  reduce  fractions  to  their  Least  Common  Denominator 
(L.  C.  D.)  is  to  express  them  as  equivalent  fractions,  having 
for  their  common  denominator  the  least  common  multiple  of 
the  given  denominators. 

93.  It  was  shown  in  Art.  87  that  the  fraction  |  is  equal 
tof. 

But  the  fraction  f  may  be  obtained  from  -|  by  multiplying 
both  numerator  and  denominator  by  2. 

Hence,  if  both  numerator  and  denominator  of  any  fraction 
be  multiplied  by  the  same  number,  the  value  of  the  fraction  is 
not  changed. 


50  ARITHMETIC. 

94.  1.  Keduce  f,  -^y  and  {^  to  their  least  cominon  de- 
nominator. 

By  Art.  73,  the  L.  C.  M.  of  6,  10,  and  15  is  30. 

Now,  by  Art.  93,  both  terms  of  a  fraction  may  be  multiplied  by  the 
same  number  without  changing  the  value  of  the  fraction. 

Multiplying  both  terms  of  f  by  5,  both  terms  of  j'^  by  3,  and  both 
terms  of  {^  by  2,  the  given  fractions  become 

2  5     21     nrirl    2  2      J,, o 

It  will  be  observed  that  the  terms  of  each  fraction  are 
multiplied  by  a  number  which  is  obtained  by  dividing  the 
least  common  denominator  by  its  own  denominator ;  hence 
the  following 

RULE. 

Mnd  the  L.  C.  M.  of  the  given  denominators. 

Divide  this  by  each  denominator  separately,  multiply  the 
respective  numerators  by  the  quotients,  and  write  the  results 
over  the  common  denominator. 

If  the  given  denominators  are  prime  to  each  other  (Art. 
64),  the  least  common  denominator  is  the  product  of  all  the 
denominators ;  and  each  numerator  is  multiplied  by  all  the 
denominators  except  its  own. 

2.   Reduce  |,  J,  and  -^  to  their  least  common  denominator. 

The  L.  C.  D.  is  3  X  4  X  5,  or  60. 

Multiplying  each  numerator  by  all  the  denominators  except  its  own, 
the  fractions  become 

fj,  Mj  and  48^  Ans. 

EXAMPLES. 

Reduce  to  their  least  common  denominator : 

3.  i,iandf  7.   A,  ^  and  ff- 

4.  I,!,  and  44.  8.   3^,  A,  and  i|. 
6.   |,f,and|.                           9.   H.  M.  and  ff. 

6.   fi,  tt  and  if.  10.   A.  ih  tt,  and  «• 


FRACTIONS.  51 

11-   A.H.tt.andf^.  13.   A.  A.  If ,  if .  and  23. 

12.   T^,  tt.  U^  and  f|.  14.   fl,  11,  il,  if,  and  ^. 

The  relative  magnitude  of  fractions  may  be  determined 
by  reducing  them,  if  necessary,  to  their  least  common  de- 
nominator. 

15.  Which  of  the  fractions,  i  and  f ,  is  the  greater  ? 

We  have,  i  =  A?  and  |  =  2^. 

It  is  evident  from  this  that  f  is  greater  than  i. 

Arrange  in  order  of  magnitude  : 

16.  ^  and  ^.  18.  f  if,  and  |f . 

17.  ^,  f,  and  |.  19.  ^,  |,  and  ^. 

95.  To  reduce  a  fraction  to  an  equivalent  fraction  having 
any  required  denominator,  divide  the  required  denominator 
by  the  given  denominator,  and  multiply  both  terms  of  the 
given  fraction  by  the  result. 

1.  Reduce  ^  to  165ths. 

Dividing  165  by  15,  the  quotient  is  11. 

Multiplying  both  terms  of  {\  by  11,  the  result  is  |fj,  Ans. 

EXAMPLES. 

2.  Reduce  y\  to  78ths.  5.  Reduce  |i  to  375ths. 

3.  Reduce  ^  to  126ths.  6.  Reduce  ff  to  504ths. 

4.  Reduce  |f  to  224ths.  7.  Reduce  |f  to  576ths. 

ADDITION  OF  FRACTIONS. 

96.  1.  "Find  the  sum  of  |,  |,  and  |. 
The  L.  C.  M.  of  3,  4,  and  6  is  12. 
Keducing  eacli  fraction  to  12ths,  we  have 


52  ARITHMETIC. 

From  the  above  example,  we  derive  the  following 

RULE. 

To  add  two  or  more  fractions,  reduce  them,  if  necessary,  to 
their  least  common  denominator. 

Add  the  numerators  of  the  resulting  fractions,  and  write  the 
result  over  the  common  denominator. 

The  final  result  should  be  reduced  to  its  lowest  terms. 

To  add  two  or  more  mixed  numbers,  first  add  the  whole 
numbers,  and  then  the  fractions,  and  then  find  the  sum  of 
these  results. 

2.  Find  the  sum  of  3J,  1^,  5,  and  2\\. 
3-1-1  +  5+2  =  11.  The  sum  of  the  whole  numbers 

1  -|_  _^  _j_  1 1  =  _5   _|_  _9_  _i_  2_2      3,  1,  5,  and  2,  is  11. 

3g g '-|j  The  sum  of  the  fractions  |^,  -^^, 

11  4_1 1-121     Aii^  ana  T^,  IS  ^,  or  i^. 

-^-^  + -^^  — -^^3-'  ^^^'  Then  the  sum  of  11  and  1|  is  12^. 

EXAMPLES. 
Find  the  values  of  the  following : 

3.  J  +  H.  6.   5H  +  3«.  9.   I  +  A  +  A- 

4.  ^  +  ^.  7.   i^  +  1^.  10.   ^^  +  ^  +  ^. 

5.  lH  +  2tt.  8.   f  +  l  +  A-  !!■   A  +  «  +  M-^ 

12.  l|  +  lf  +  l|.  21.  ll^  +  21H  +  1444..^7i^ 

13.  1^  +  21  +  3,%.  22.  4  +  3|  +  2A  +  l^.    /;  ^ 

14.  7J  +  3f  +  5f.  l'(     23.  2 +  61  +  81  + 4A.     >l  >^^ 

15.  f  +  f  +  ^  +  T^j.  a,.     24.  ^  +  ^  +  ^  +  l^.     ,j,V^ 

16.  A  +  A  +  iV  +  A-  .  -.'25.  6i  +  5i  +  4|  +  3A.       ?  ;J 

17.  A  +  T^  +  M  +  «-  ^  -»   26.  3i  +  9i  +  7|  +  5A.  .,^»^Vt^ 
18-  i  +  |  +  A  +  /T-  27.  2A  +  4A  +  6^  +  8||.  vc'-jj 

19.  6ii  +  4||+8|i.  28.   J  +  l  +  l  +  l  +  J.     l:^ 

20.  4^  +  9H  + 13^.  29.  A  +  A  + 1^  +  A  +  A-  '^l^ 

■3    ^l' 


FRACTIONS.  -^     53 

30.  7f4-9/^  +  2A  +  12fi  +  34|. 

31.  13i  +  16|  + 191  +  22,7^  +  25^3. 

32.  34i  + 171  +  28^3^  + 40,^ +  52^.       )  7>  ^b^ 

SUBTRACTION  OF  FRACTIONS. 
97.   1.    Subtract  y\  from  f 
Reducing  each  fraction  to  42(is,  we  have 

5. 9_  —  SA  2  7  —     8     4  y(„  o 

6  1  4   —  42^  4  2  —  ¥2"  —  2"TJ    ^***- 

From  the  above  example,  we  derive  the  following 

RULE. 

To  subtract  one  fraction  from  another,  reduce  them,  if  neces- 
sary, to  their  least  common  denominator. 

Subtract  the  numerator  of  the  subtrahend  from  that  of  the 
minuend,  and  write  the  result  over  the  common  deyiominator. 

The  j&nal  result  should  be  reduced  to  its  lowest  terms. 

To  subtract  one  mixed  number  from  another,  first  sub- 
tract the  integers,  and  then  the  fractions,  and  then  find  the 
sum  of  these  results. 

2.  Subtract  3|-  from  5|. 

^  ~  3  =  2.  Subtracting  3  from  5,  the  result  is  2. 

|-  —  |-  =  \^  —  T g"  =  T^'  Subtracting  |  from  f ,  the  result  is  y\. 

2  +  ^  =  2^,  Ans.  Then  the  sum  of  2  and  ^\  is  2^^- 

If  the  fractional  part  of  the  subtrahend  exceeds  the  frac- 
tional part  of  the  minuend,  increase  the  latter  by  1,  subtract- 
ing 1  from  the  integral  part  of  the  minuend  to  compensate. 

3.  Subtract  3|  from  5f 

.       Q  _  i  ^  Since  f  is  greater  than  |,  we  subtract  1 

~     ~    *  from  5,  leaving  4,  and  then  add  f  to  the  \, 

■6~¥~  is^TF^  T8-  giving  I ;  thus,  5^  is  the  same  as  4|. 
IW,  Ans. 


64  ARITHMETIC. 


4.  Subtract  2f  from  7. 

7-2|  =  6|-2|  =  4|,^n5. 


EXAMPLES. 


5. 

Lu  uiit;  values 
T2~T(J- 

Ul    UJf 

14. 

!  iUiiuwiiig  ; 

23. 

18A-8«- 

7 

7; 

6. 

i^—h- 

15. 

li-A- 

24. 

16if-9|i- 

7. 

A-1%- 

16. 

6T~Tg"* 

26. 

23if-17H. 

^'3^- 

8. 

2-tV- 

17. 

M-H- 

26. 

12i^-if 

9. 

5-3tf. 

18. 

8i^-2A. 

27. 

33A-25A. 

-r-  y 

10. 

If-A- 

19. 

7tt-6,V 

28. 

27A-19H. 

11. 

9-H- 

20. 

lA-l*- 

29. 

17if-4H. 

12. 

11-7M. 

21. 

12i*-5tt- 

30. 

24A-15tt- 

13. 

A-A- 

22. 

UJI-lOi^. 

31. 

31A-18«- 

In  finding  the  value  of  a  series  of  fractions  connected  by 
plus  and  minus  signs,  it  is  better  to  add  all  those  fractions 
which  are  preceded  by  minus  signs,  and  subtract  their  sum 
from  the  sum  of  the  other  fractions. 

32.  Find  the  value  of  4iJ  -  3 ^l  +  2||  -  1 1|. 

%  +  ltt  =  4^Wi  =  4|| 

2f|  =  2H,  Ans. 

The  sum  of  the  fractions  4||  and  2||  is  6f f . 
The  sum  of  the  fractions  3y\  and  \\\  is  4||. 
Subtracting  4|f  from  6f |,  the  result  is  2f f . 

Find  the  values  of  the  following : 

33.  5J-3i-2J,. 

34.  4i-li  +  8|-7i. 

37.  9ii- 


35.  91  +  2^-5,2,- 

-3f 

36.  l^-\\-ll. 

-If 

3if  +  7f  +  A-6i. 

FRACTIONS.  55 

38.  253L  4. 20f  - 1711  -  IS^V  -  8^. 

39.  52| -15^-31- 2611  + 9f|. 

40.  12H-6A-3H  +  7ti-H  +  8^. 

41.  S2^  + 17/3  -  18,^  -  U^  +  20H  -  lOif. 

42.  2113  -  11 J3  -  5ff  +  58||  -  12e  +  t\\  -  19if 

MULTIPLICATION  OF  FRACTIONS. 

98.  To  Multiply  a  Fraction  by  a  Whole  Number. 

Let  the  line  AD,  in  the  following  figure,  represent  one 
unit ;  and  let  it  be  divided  into  8  equal  parts. 


f  or  f  1 

Then  the  fraction  |-  will  be  represented  by  AB,  and  f  by 
AG. 

But  AC  is  twice  AB ;  hence,  the  fraction  |  is  twice  f . 

Now  f  may  be  obtained  from  |^  by  multiplying  its  numer- 
ator by  2 ;  hence,  if  the  numerator  of  a  fraction  be  multiplied 
by  any  number,  the  fraction  is  multiplied  by  that  number. 

Again,  since  f  is  equal  to  f ,  the  fraction  f  is  twice  f. 

But  f  may  be  obtained  from  |  by  dividing  its  denomina- 
tor by  2;  hence,  if  the  denominator  of  a  fraction  be  divided 
by  any  number,  the  fraction  is  multijMed  by  that  number. 

99.  We  derive  from  Art.  98  the  following  rule  for  mul- 
tiplying a  fraction  by  a  whole  number  : 

If  possible,  divide  the  denominator  by  the  whole  number; 
otherwise,  multiply  the  numerator  by  the  whole  number. 

1.    Multiply  3^  by  5. 

Dividing  the  denominator  by  5,  we  have 

-53^X0  =  1,  Ans. 


56  ARITHMETIC. 

2.  Multiply  f  by  4. 

Multiplying  the  numerator  by  4,  we  have 

1x4  =  \%  Ans. 

Common  factors  in  the  whole  number  and  the  denomina- 
tor of  the  fraction  should  be  cancelled  (Art.  90)  before  per- 
forming the  multiplication. 

3.  Multiply  If  by  18. 

2 
1^  V  tc_  1^  V  9      2^     A^o         111  this  case,  we  cancel  9  from 

3 

Note.    To  multiply  a  whole  number  by  a  fraction  is  the  same  as 
multiplying  the  fraction  by  the  whole  number. 
Thus,  5  X  f  is  the  same  as  |  x  5. 

To  multiply  a  mixed  number  by  an  integer,  multiply  the 
whole  number  and  the  fraction  separately,  and  then  find 
the  sum  of  these  results. 

4.  Multiply  3i|  by  12. 

3  X  12  =  36. 

2  Multiplying  3  by  12,  the  product  is  36. 

li  y  791  =  —  =81  Multiplying  \\  by  12,  the  product  is  ^^-^ 

^"^^^       4         ^*  or  8^. 

4  Then  the  sum  of  36  and  8J  is  44J. 
36-f  8^  =  44^,  Ans. 

EXAMPLES. 
Find  the  values  of  the  following : 

5.  If  X  9.        9.  117  X  \%.    13.  42=V  X  3.  17.  15-\  x  18. 

6.  -A-  X  8.       10.  IJ  X  80.       14.  82  X  6.  18.  35  x  17-^:. 

7.  ^  X  42.     11.  66  X  \\.       15.  9  x  7J^.  19.  25  x  22^^. 

8.  75  X  i|.     12.  ^  X  72.       16.  14^  x  33.  20.  16|  x  64. 

21.  80  X  18|^.  22.  30|f  x  84. 


FRACTIONS.  57 

100.  To  Multiply  a  Fraction  by  a  Fraction. 

To  multiply  |  by  f  is  to  take  |  0/  |  j  that  is,  we  divide  f 
into  5  equal  parts,  and  take  4  of  them. 

Let  the  line  AC,  in  the  following  figure,  represent  one 
unit ;  and  let  it  be  divided  into  15  equal  parts. 

A  B  E  F  G  B  G 


I  of  \f        ^  1 

Then  AB  will  represent  if,  or  |. 

Now  since  ^5  is  divided  into  5  equal  parts,  AD,  DE,  EF, 
FG,  and  GB,  AG  will  represent  |  of  |. 
But  AG  also  represents  -^-^. 
Hence,  f  of  |,  or  |  x  |,  is  equal  to  y%. 

101.  We  derive  from  Art.  100   the  following  rule  for 
multiplying  one  fraction  by  another  : 

Multiply  the  numerators  together  for  the  numerator  of  the 
product,  and  the  denominators  for  its  denominator. 

Common   factors   in  the  numerators   and   denominators 
should  be  cancelled  before  performing  the  multiplication. 


1.  Multiply!  by 


^x^='^,Ans. 


To  multiply  any  number  of  fractions,  we  multiply  their 
numerators  together  for  the  numerator  of  the  product,  and 
their  denominators  for  its  denominator. 

2.  Find  the  value  of  f  x  |  of  \\. 

2 

0^11_22    ^  In  this  case,  we  cancel  3  from  6  and  9, 

7      ^  ^  "  63'  *        and  then  5  from  5  and  15. 

3        3 

Mixed  numbers  should  be  reduced  to  a  fractional  form 
(Art.  85)  before  applying  the  rule. 


58  ARITHMETIC. 

3.  Find  the  value  of  ||  of  1 J^  x  2^  x  9. 

If  of  1^^  X  2^2^  X  9  By  Art.  85,  1^\  is  equal  to  f^, 

5         9        2      3  and  2t?5  toff. 

cm      of»      QO              OT-  ^^®  ^^^*  cancel  16  from  48  and 

=  fE  X  ^  X^  X  ^  =  —,  Ans.    32  ;  then  5  from  25  and  15  ;  then 

^      ^^      X^               2  5  from  5  and  20 ;   then  3  from  3 

^        ^        ^  and  27  ;  then  2  from  2  and  4 ;  and 

2  finally  3  from  3  and  9. 

EXAMPLES. 
Find  the  values  of  the  following : 
4-  Ax  J.  6.  lHx2Jf.  8.  ^of-V/. 

6.  «  X  2^.        7.  if  of  2^.  9-  3tt  x  IM- 

10.  if  X  H  X  «.     _  16.  I  X  A  X  if  X  28.    -> 

11.  li  X  H  of  5f.     '  _  ,  17.  if  of  ^\  of  26  X  4^.  :- 

12.  ifxi|x5f.  _       18.  ^o^xMxIfof^.    Jt; 

13.  if  of  4,V  X  2|f . .    ■  19.  ii  of  If  of  li  of  T%. 

14-  T*A  of  U  of  ff •  20.  t7jV  X  A  x  ifi  X  35.    3> 

15.  lA  X  2A  x  1^.  \  21.  Ifi  x  l|f  x  2H  X  in-^ 

22-  A  x'^  X  20  X  2^  X  lOif.  ^ 

23.  il  of  ^  of  2if  X  24  X  3^.     j^ 

•"-^ 

DIVISION  OF  FRACTIONS. 

102.  To  Divide  a  Fraction  by  a  Whole  Number. 

B  .         c  D 


i  or  I  1 

It  is  evident  from  the  above  figure  that  tlie  fraction  |.  is 
equal  to  the  fraction  ^  divided  by  2. 

But  f  may  be  obtained  from  |  by  dividing  its  numerator 
by  2. 


FRACTIONS.  59 

Hence,  if  the  numerator  of  a  fraction  he  divided  by  any 
number,  the  fraction  is  divided  by  that  number. 

Again,  the  fraction  f  is  equal  to  f  divided  by  2. 

But  I  may  be  obtained  from  J  by  multiplying  its  denom- 
inator by  2;  hence,  if  the  denominator  of  a  fraction  be  multi- 
plied by  any  number,  thefractioyi  is  divided  by  that  number'. 

103.  We  derive  from  Art.  102  the  following  rule  for 
dividing  a  fraction  by  a  whole  number  : 

If  possible,  divide  the  numerator  by  the  whole  number; 
otherwise,  multiply  the  denominator  by  the  whole  number. 

1.  Divide  f  by  3. 

Dividing  the  numerator  by  3,  we  have  . 

f-r-3  =  f,  Ans. 

2.  Divide  |  by  5. 

Multiplying  the  denominator  by  5,  we  have 

To  divide  a  mixed  number  by  an  integer,  the  dividend 
should  first  be  reduced  to  a  fractional  form  (Art.  85). 

3.  Divide  ^  by  8. 

By  Art.  85,  6|  =  V- ;  and  ^-  ^  8  =  |,  Ans. 
If  the  integral  part  of  the  mixed  number  is  equal  to  or 
greater  than  the  divisor,  it  is  better  to  proceed  as  follows : 

4.  Divide  86f  by  12. 

19^Sfi3  ^^  ^^  contained  in  86f  seven  times,  with  a  re- 

^    J*  mainder  of  2f ,  or  J/-. 

1^,  Ans.  Dividing  -V-  by  12,  the  quotient  is  ^f 

EXAMPLES. 
Find  the  values  of  the  following : 

5.  A2^7.  8.^-4-9.        11.   2H-6.     14.401-^11. 

6.  i|-^5.  9.   l2%-^2.        12.   16|-^7.     15.  9H-4. 

7.  \3^-f-12.    10.   83^-^10.       13.   lOi-5-3.     16.29^-^8. 


60  ARITHMETIC. 

104.  To  Divide  a  Whole  Number  or  a  Fraction  by  a 
Fraction. 

1.  Divide  3  by  f 

We  have,  3  -  f  =  -V"  -^  I- 

But  the  quotient  of  21  sevenths  divided  by  5  sevenths  is  the  same  as 
the  quotient  of  21  divided  by  5,  which  is  V- 
Therefore,  3  ^  f  =  V.  -^^i*- 

We  observe,  in  the  above  example,  that  the  quotient  may 
be  obtained  by  multiplying  3  by  ^,  which  is  the  fraction  ^ 
inverted;  whence  the  following 

RULE. 

To  divide  a  whole  number  or  a  fraction  by  a  fraction,  invert 
the  divisor,  and  proceed  as  in  multiplication. 

2.  Divide  ^  by  |. 

9    _i_8  —     9     v9  —  81       Art'i 

If  the  numerator  and  denominator  of  the  divisor  are  ex- 
actly contained  in  the  numerator  and  denominator  of  the 
dividend,  we  divide  the  numerator  of  the  dividend  by  that 
of  the  divisor  for  the  numerator  of  the  quotient,  and  the 
denominator  of  the  dividend  by  that  of  the  divisor  for  the 
denominator  of  the  quotient. 

3.  Divide  ff  by  ^. 

Since  35  -=-  7  =  5,  and  44  h-  11  =  4,  we  have 

If  the  divisor  is  an  integer,  it  must  be  written  in  a  frac- 
tional form  (Art.  81)  before  applying  the  rule. 

4.  Divide  f|  by  63. 

4 
36_^63__^1  By  Art.  81,  63  may  be  written  in  the  form 

25  *    1        25       0^  ^T^- ;  which,  when  inverted,  becomes  ■^^. 
7  We  cancel  9  from  36  and  63. 


FRACTIONS.  61 

If  either  the  dividend  or  divisor  is  a  mixed  number,  it 
must  be  expressed  in  a  fractional  form  before  applying  the 
rule. 

5.  Divide  21  by  3{\. 

21  -^  3||  =  21  -7-  f  f  ^1  is  tije  same  as  ||. 

3       ^  ^       ^  ^  We  cancel  7  from  21  and  56. 

8 

EXAMPLES. 

Find  the  values  of  the  following : 


6-  n-^H- 

14. 

V-2f. 

22.  18^  +  4«. 

7.  27-f-i^. 

15. 

fl^A- 

23.  fl  +  if. 

8.  Hh-100. 

16. 

28  ^«. 

24.  ^A»-.^51^. 

9.  45H-6J,-. 

17. 

3H  ^  96. 

25.  Hf^^. 

10.  ll^lSj^j. 

18. 

lH^7i| 

26.  Itt^l^j. 

U.  3tV-2^. 

19. 

3»-fi. 

27.  T%^lif. 

12.  2^15.^12. 

20. 

W  :  88. 

28.  W-W- 

13.  iH-¥- 

21. 

85^  Iff. 

29.  2TS^^2i|. 

30.  n^(i-i). 

35.  (|H 

-|)  +  (i^2T^). 

31.  n^axfi). 

36.  (« 

0ffi)x(if+ff). 

32.  (iofT7j)x(|- 

Ht). 

37.  a- 

-|)  +  (Y  +  A)- 

33.  (A  +  A)-lf 

38.  (|ofl}xH)-ll- 

34.  (|  +  i)^(f- 

f). 

39.  (If 

^4J,)  +  (ii  +  2T^) 

COMPLEX  FRACTIONS. 

105.  A  Complex  Fraction  is  one  having  a  fraction  in  its 
numerator  or  denominator,  or  in  both. 

A  fraction  both  of  whose  terms  are  integers  is  called  a 
Simple  Fraction. 


62  ARITHMETIC- 

A  complex  fraction  may  be  regarded  as  a  case  in  divi- 
sion of  fractions;  and  it  may  be  reduced  to  a  simple 
fraction  by  the  rule  of  Art.  104. 

5 

1.  Reduce  ^  to  a  simple  fraction. 

-^  is  the  same  as  f -^-^ ;  inverting  the  denominator,  we  have 
"^5  3 

10  =  p^^  =  4'^''^- 


Another  method  is  to  multiply  both  numerator  and  denomi- 
nator by  the  least  common  multiple  of  their  denominators. 

Thus  Ex.  1  may  be  solved  as  follows : 

The  L.C.M.  of  6  and  9  is  18  ;  then  multiplying  both  numerator  and 
denominator  by  18  (Art.  93),  we  have 


5 
6 

1^ 

18 

15 

3       A 

10" 
9 

i"^ 

18 

20~ 

4'^' 

2. 

Simplify 

2^. 
42 

63 

3 

• 

42 

29 
~42' 
1 

"29 

1 

2 

3 

=  58' 

3. 

SimDlifv 

3i  +  H 

'     •'   5f-3f 

The  L.  C.  M.  of  the  denominators  3,  8,  6,  and  4,  is  24. 
Multiplying  each  term  of  the  fraction  by  24,  we  have 

80  +  45       125      5      . 

-        =-,  Ans. 


140  -  90       50       2 


FRACTIONS.  63 


EXAMPLES. 

Simplify  the  following : 


'  S-     '■ 

56 

10. 

51 

1 1 

13.  m 

«•  1    «• 

H 

n 

11. 

IH 

6A 

14.    «. 

«•  1-     «• 

2^ 

12. 

if. 

15.  T%. 

2fJ 

,6.   t  +  ¥. 

17. 

5*- 
4|- 

-31 

18.  2^  +  li 

8-5| 

19.     ^°ft. 
ffof2f 

22. 

6i.-4i  +  2| 
7i  +  3A-8J 

20    lA..^. 
•   2A  •  lOi 

23. 

3H  +  2f  +  5T^ 
10A-lf-3f 

21     foff-T\ofJ^ 
fofA  +  ttofil 

24. 

TO"  ~~  TO  +  "60"  ~  Tl" 

H  +  tt-fV-A 

25.   A' 

ofW'  +  l 
°f2TV-f 

ofl^ 
ofyV 

-• 

106.  The  Reciprocal  of  a  number  is  1  divided  by  t^hat 
number. 

Thus,  the  reciprocal  of  5  is  ^. 

The  reciprocal  of  |-  is  -,  or  ^. 

"g" 
That  is,  the  reciprocal  of  a  fraction  is  the  fraction  inverted. 

107.  To  find  what  Fraction  one  Number  is  of  another. 
1.  What  fraction  of  21  is  14  ? 

Since  1  is  Jy  of  21,  14  is  14  times  ^\  of  21,  or  ii  of  21. 

Result,  if,  or  |. 

From  the  above  example,  we  derive  the  following 

RULE. 

Make  the  first  number  the  denominator^  and  the  second  the 
numerator,  of  a  fraction. 


64  ARITHMETIC. 

EXAMPLES. 

What  fraction  of : 

2.  36  is  27  ?  7.  6i  is  11  j  ?  12.  2^  is  2-jV  ? 

3.  49  is  70  ?  8.  2^  is  24  ?  13.  90  is  4:j\  ? 

4.  IJ  is  H?  9-  40  is  5^^?  14.  H  is  IM? 

5.  41-  is  3f  ?  10.  ff  is  II  ?  15.  13if  is  68  ? 

6.  H  is  if?  11.  43V  is  7f  ?  16.  1^  is  Iff? 

17.  |  +  |is|  +  ^?  21.  f  ofiis|xl|? 

18.  9  +  4iisll-3i?  22.  I|x2|is  fof  31? 

19.  5f-2f  is4A_|-i|?  23.  6i  +  3|  is  5^2-21? 

20.  6|-2|  is  5I-41-?  24.  if  of  2^  is  ^  of  1|? 

25.  f+A-«is|  +  f  +  i? 

26.  3|-l|  +  5His53^  +  2|-3|f? 

108.  To  find  a  Number  when  one  of  its  Fractional  Parts  is 
given. 

1.  7  is  f  of  what  number  ? 

K  7  is  I,  one-ninth  of  the  required  number  will  be  i  of  7,  or  |. 
Then  the  required  number  is  9  times  |,  or  -*'/,  Ans. 

It  is  evident  from  the  above  that  the  required  result  may- 
be obtained  by  multiplying  the  first  number  by  the  second 
number  inverted. 

2.  2iV  is  f  of  what  number  ? 

5      3 

s 

EXAMPLES. 

3.  8  is  If  of  what  number  ? 

4.  28  is  |-J  of  what  number  ? 


FRACTIONS.  65 

5.  -^  is  f§-  of  what  number  ? 

6.  Sj\  is  fl  of  what  number  ? 

7.  f^  is  y\  of  what  number  ? 

8.  1^  is  ff  of  what  number  ? 

9.  f|-  is  II  of  what  number  ? 

10.   I  of  i|  is  ^  of  what  number  ? 

GREATEST  COMMON  DIVISOR  OF  FRACTIONS. 

109.  The  Greatest  Common  Divisor  of  two  or  more  frac- 
tions is  the  greatest  fraction  that  is  contained  in  each  of 
them  an  integral  number  of  times. 

In  order  that  one  fraction  may  be  contained  in  another 
an  integral  number  of  times,  its  numerator  must  be  a  divisor 
of  the  numerator,  and  its  denominator  a  multiple  of  the 
denominator,  of  the  second  fraction. 

Thus,  I  is  contained  an  integral  number  of  times  in  |, 
since  2  is  a  divisor  of  4,  and  9  a  multiple  of  3. 

Now,  the  greater  the  numerator  of  a  fraction,  and  the 
smaller  its  denominator,  the  greater  is  the  value  of  the 
fraction. 

Hence,  the  greatest  common  divisor  of  two  or  more  frac- 
tions is  the  greatest  common  divisor  of  their  numerators^ 
divided  by  the  least  common  multiple  of  their  denominators. 

1.  Find  the  G.  C.  D.  of  -^,  |f,  and  ^<>. 

The  G.  C.  D.  of  24,  16,  and  40,  is  8. 
The  L.  C.  M.  of  5,  15,  and  9,  is  45. 
Then,  the  required  G.  C.  D.  is  ^r,^  Ans. 

EXAMPLES. 

Find  the  G.  C.  D.  of: 

2.  A.  M.  and  ^.  6.  33f ,  94|,  and  37i. 

3.  41.,  4^^,  and  6^.        7.  |,  if,  4f ,  and  13f 

4.  ^,  1^,  and  f|.  8.  73^,  1248,  and  39||. 

5.  2^,  fi,  and  2^.         9.  |4,  |f ,  ^,  and  f|. 


66  ARITHMETIC. 

LEAST  COMMON  MULTIPLE  OF  FRACTIONS. 

110.  The  Least  Common  Multiple  of  two  or  more  frac- 
tions is  the  smallest  number  that  will  contain  each  oi  them 
an  integral  number  of  times. 

In  order  that  one  fraction  may  contain  another  an  inte- 
gral number  of  times,  its  numerator  must  be  a  multiple  of 
the  numerator,  and  its  denominator  a  divisor  of  the  denom- 
inator, of  the  second  fraction. 

Thus,  f  contains  f  an  integral  number  of  times,  since  4  is 
a  multiple  of  2,  and  3  a  divisor  of  9. 

Now,  the  smaller  the  numerator  of  a  fraction,  and  the 
greater  its  denominator,  the  smaller  is  the  value  of  the 
fraction. 

Hence,  the  least  common  multiple  of  two  or  more  fractions 
is  the  least  common  multiple  of  their  numerators,  divided  by 
the  greatest  common  divisor  of  their  denominators, 

1.  Find  the  L.  C.  M.  of  ^,  ^j,  and  ^. 

The  L.  C.  M.  of  3,  4,  and  9,  is  36. 
The  G.  C.  D.  of  14,  21,  and  35,  is  7. 
Then,  the  required  L.  C.  M.  is  -^^,  Ans. 


Find  the  L.  C.  M.  of : 

EXAMPLES. 

\ 

2.  1,  U,  and  If 

3.  2|,  If,  and  3f 

4.  T-\,  f ,  and  ,V 

5.  ^,  ii,  and  2^. 

6.  25.V,A,and^. 

7.  2f ,  3i|,  and  3^. 

8-  A.  A>  A>  and  ^. 

9-  ^A,T¥r.T¥r,an'i*k- 

MISCELLANEOUS    EXAMPLES. 
111.  1.  Reduce  -^f^  to  a  mixed  number. 

2.  Reduce  28  to  a  fraction  having  137  for  a  denominator. 

3.  Multiply  2^  by  18.  5.  Divide  4f  by  11. 

4.  Multiply  m  by  23.  6.  Divide  lif  by  19. 


FRACTIONS.  67 

7.  Reduce  123f  |  to  an  improper  fraction. 

8.  Arrange  in  order  of  magnitude  fj,  ||,  and  |f . 

9.  Reduce  ff  to  768tlis. 

10.  Divide  33^^  by  12. 

11.  Divide  37^V  ^J  ^i- 

12.  J^  of  ^  is  f  I  of  what  number  ? 

13.  Reduce  |-fi-ff  to  its  lowest  terms. 

14.  Find  the  value  of  3i  X  5^  x  7^  x  OJ. 

15.  Reduce  -VAV"  *^  ^*^  lowest  terms. 

16.  Add  together  2j\,  5\i,  8if ,  and  llif . 

17.  Find  the  value  of  (2  -  fi)  X  (2  -  fj). 

18.  Subtract  ^j9^  from  Iff . 

19.  Find  the  value  of  51  -f-  (3  —  ^) . 

20.  Find  the  value  off-f  +  i-f  +  f     • 

21.  Divide  2j^  by  l^^,.. 

22.  Add  together  J^,  |f ,  {i,  and  if^. 

23.  What  fraction  of  ^  of  2i  is  2^^  ? 

24.  Reduce  f |ff  to  its  lowest  terms. 

25.  Subtract  13i|  from  23if . 

26.  Multiply  together  |f ,  f  §,  2-^,  and  2^. 

27.  Find  the  value  of  (3  -  ^)  -^  (1  - 1%). 

28.  Add  together  5|,  6|,  7f ,  8f ,  and  9f . 

29.  Arrange  in  order  of  magnitude  ^,  If,  ||. 

30.  Multiply  together  ^^  2^%  m,  and  1^^^. 

31.  Find  the  G.  C.  D.  of  2f  |,  4^^  5^,  and  9f 

32.  Find  the  L.  C.  M.  of  ^,  H,  H,  ff ,  and  ff . 

33.  Simplify  ^^^. 

34.  Find  the  value  of  (3^  +  4|)  -  (2^  +  1  A)- 
„^    ^.      ....      539x637x66 

3^-  ^^^P^^^^  34^^^23r^^^* 


68  ARITHMETIC. 

36.  Simplify  ^-H  +  H-H 

37.  Simplify  (|  of  2^)  +  (f  of  8^)  -  (|  of  2^)  -  (f  of  1|). 

38.  Simplify  i^  of  i?  -  §^  +  ^  of  3f . 

39.  Simplify -^^^jy-^j^^^Q^. 

40.  Simplify  ||^  +  (|  of  3f )  -  (1/^  ^  H)  -  fi- 

41.  Simplify  (4AxlH)-(3A^5i). 

42.  Simplify  (H  X  W)  +  (H  X_ii) . 

VT¥    •    6  3/        ^16    •    TUf 


PROBLEMS. 

112.  1.  If  a  man  can  do  f  of  a  piece  of  work  in  2^ 
hours,  in  how  many  hours  can  he  do  the  whole  ? 

If  he  can  do  five-ninths  of  the  work  in  f  f  hours,  he  can  do  one- 
ninth  in  one-fifth  of  f  f  hours ;  and  he  can  do  nine-ninths,  or  the 
whole,  in  9  times  i,  or  f  of  f  f  hours. 

3      5 

?  X  ^  =  —  =  3|  hours,  A71S. 
^     X^      4 
4 

2.  A  tank  can  be  filled  by  one  pipe  in  8  minutes,  and  by 
another  in  12  minutes.  How  many  minutes  will  it  take  to 
fill  the  tank,  if  both  pipes  are  opened  ? 

The  first  pipe  in  one  minute  will  fill  I  of  the  tank,  and  the  second 
in  one  minute  will  fill  ^2  of  the  tank. 

Then  both  together  will  fill  i  +  tV»  o^  2T  of  the  tank  in  one  minute. 

Then  it  will  take  as  many  minutes  for  both  pipes  together  to  fill  the 
tank  as  /^  is  contained  times  in  f| ;  that  is  ^-^%  or  4|  minutes,  Ans. 


FRACTIONS.  69 

3.  If  4|  tons  of  coal  is  worth  $  311,  how  much  is  7^  tons 
worth  ? 

If  y-  tons  is  worth  ^-  dollars,  one  ton  is  worth  as  many  dollars  as 
-Li  is  contained  times  inA^. 

9 

2       3       2      ;^      4' 
2 

Then,  if  one  ton  is  worth  ^^  dollars,  -y-  tons  will  be  worth  y^  times 
^  dollars. 

16      3 

9       i 

4.  A  can  do  a  piece  of  work  in  12  days,  B  can  do  the 
same  work  in  14  days,  and  C  in  21  days.  How  many  days 
will  it  take  all  of  them  together  to  do  the  work  ? 

5.  A  tank  can  be  emptied  by  one  pipe  in  9f  minutes,  and 
by  another  in  lOf  minutes.  How  many  minutes  will  it 
take  to  empty  the  tank  if  both  pipes  are  opened  ? 

6.  A  man  walked  63  miles.  He  performed  the  first  half 
of  his  journey  at  the  rate  of  4^  miles  an  hour,  and  the  last 
half  at  the  rate  of  5^  miles  an  hour.  How  many  hours  did 
it  take  him  ? 

7.  If  3^  of  a  ton  of  hay  is  worth  $8|-,  how  much  is  10 
tons  worth  ? 

8.  A  man  having  lost  ^  of  his  money,  and  then  spent  y% 
of  the  remainder,  found  that  he  had  $  112  left.  How  much 
had  he  at  first  ? 

9.  How  many  pecks  of  apples,  at  25J  cents  a  peck,  must 
be  given  for  12|^  pounds  of  sugar,  at  4f  cents  a  pound  ? 

10.  I  sold  a  house  and  lot  for  $  3125,  which  was  |f  of 
what  they  cost  me.  How  much  did  I  lose  by  the  opera- 
tion? 

11.  The  circumference  of  the  hind- wheel  of  a  carriage  is 
9f  feet,  and  of  the  fore-wheel  8f  feet.  How  many  times 
does  each  wheel  turn  in  travelling  5280  feet  ? 


70  ARITHMETIC. 

12.  A  merchant  who  owned  J  of  a  ship,  sold  |^  of  his 
share  for  f  15625.  What  was  the  value  of  the  whole  ship 
at  the  same  rate  ? 

13.  A  man  sold  a  horse  and  carriage  for  $  624,  receiving 
f  as  much  for  the  horse  as  for  the  carriage.  What  did  he 
receive  for  each  ? 

■    14.  If  a  horse  travels  7|-  miles  an  hour,  how  long  will  it 
take  him  to  travel  20|-  miles  ? 

15.  If  3-j^  is  |-|  of  a  certain  number,  what  is  J|-  of  the 
same  number  ? 

16.  What  number  must  be  multiplied  by  3|^,  so  that  the 
product  may  be  20|-  ? 

17.  A  man  spent  ^  of  his  money,  and  then  received  $105, 
when  he  found  that  he  had  f  of  his  original  amount.  How 
much  had  he  at  first  ? 

18.  My  income  is  $  8f  a  week,  and  my  expenses  are  $  5^j 
a  week.     How  many  weeks  will  it  take  me  to  save  $  lOOi  ? 

19.  A  bale  of  cloth  contains  75  pieces,  each  piece  contain- 
ing 23|-  yards.     What  is  the  whole  worth  at  $  If  a  yard  ? 

20.  What  number  is  that  if  of  which  exceeds  ^^  of  it  by 
111? 

21.  If  j^  of  a  piece  of  land  is  worth  $604|,  how  much  is 
ii  of  it  worth  ? 

22.  A  dealer  has  58f  tons  of  coal  in  his  yard.  On  each  of 
six  successive  days  he  puts  in  9|-  tons,  and  sells  on  each 
day  5|  tons.  How  many  tons  has  he  in  his  yard  at  the  end 
of  the  sixth  day  ? 

23.  If  a  rod  2  feet  long  casts  a  shadow  |  of  a  foot  long  at 
12  o'clock,  how  high  is  a  flag-pole  which  casts  a  shadow  35| 
feet  long  at  the  same  time  ? 

24.  If  3f  pounds  of  sugar  cost  18  cents,  how  much  will 
6|  pounds  cost  ? 


FRACTIONS.  71 

25.  If  a  town  pays  $  480  for  the  supi)ort  of  14  paupers 
for  15  weeks,  how  much  should  it  pay  for  the  support  of  25 
paupers  for  21  weeks  ? 

26.  A,  B,  and  C  found  a  purse  containing  money.  A  took 
^  of  the  money ;  B  then  took  |  of  what  remained,  and  C  ^ 
the  remainder,  which  was  $  10|^.  How  much  money  did  the 
purse  contain  ? 

27.  In  a  certain  school,  -^  of  the  pupils  are  in  the  fourth 
class,  ^  in  the  third  class,  ^  in  the  second  class,  and  the 
remainder,  27,  in  the  first  class.  How  many  pupils  are  there 
in  each  class  ? 

28.  I  have  three  fields  containing,  respectively,  5-|  acres, 
4^^  acres,  and  111-  acres.  Find  the  size  of  the  largest  house 
lots,  all  of  the  same  size,  into  which  the  fields  can  be 
divided. 

29.  If  8f  tons  of  coal  can  be  bought  for  $  37|,  how  many 
tons  can  be  bought  for  $  22f  ? 

30.  If  a  man  can  walk  26|-  miles  in  6^  hours,  how  far  can 
he  walk  in  8^  hours  ? 

31.  A  merchant  sold  goods  for  $451,  and  gained  f  of 
what  they  cost  him.  How  much  did  he  gain  by  the  opera- 
tion? 

32.  If  20f  acres  of  land  cost  $  8000,  how  much  will  13J 
acres  cost  ? 

33:  If  a  man  can  do  a  piece  of  work  in  7^  days,  working 
11|  hours  a  day,  how  many  days  will  it  take  him  working 
9^  hours  a  day  ? 

34.  A  tank  has  two  pipes.  One  fills  it  at  the  rate  of  13^ 
gallons  an  hour,  and  the  other  discharges  the  contents  at 
the  rate  of  5|  gallons  an  hour.  If  the  tank  holds  18|^  gallons, 
how  many  hours  will  it  take  to  fill  it  ? 

35.  A  can  mow  a  field  in  5  days,  and  A  and  B  together 
can  mow  it  in  3^  days.  How  many  days  will  it  take  B  alone 
to  mow  the  field  ? 


72  ARITHMETIC. 

36.  I  have  $39  in  the  bank.  If  my  income  is  $7f  a 
week,  and  my  expenses  $  9^  a  week,  how  many  weeks  will 
my  fund  last  me  ? 

37.  If  a  man  can  do  a  piece  of  work  in  If^  days,  what 
part  of  it  can  he  do  in  ly^^  days  ? 

38.  If  the  dividend  is  f  of  21|,  and  the  quotient  |  of  6J, 
what  is  the  divisor  ? 

39.  If  a  man  can  do  a  piece  of  work  in  5f  days,  working 
8^  hours  a  day,  how  long  will  it  take  him  working  9f  hours 
a  day  ? 

40.  A  dealer  bought  a  number  of  bales  of  silk,  each  con- 
taining 135  yards,  at  $  If  a  yard,  and  sold  it  at  $  2^  a  yard, 
gaining  $  792  by  the  transaction.  How  many  bales  did  he 
buy? 

41.  Two  pendulums  beat  once  in  |-|  of  a  second,  and  once 
in  If  of  a  second,  respectively.  If  at  any  time  the  beats 
occur  together,  after  how  many  seconds  will  they  again 
occur  together  ? 

42.  The  product  of  three  numbers  is  1|| ;  if  two  of  them 
are  1^  and  2-^-^,  what  is  the  third  ? 

-  43.  A  can  do  a  piece  of  work  in  15  hours,  B  in  20  hours, 
and  C  in  30  hours.  B  and  C  worked  alone  for  5  hours,  when 
A  joined  them.  How  many  hours  will  it  take  all  of  them 
together  to  hnish  the  work  ? 

44.  If  a  man  travels  3i  miles  an  hour,  and  9^  hours  a 
day,  how  many  days  will  it  take  him  to  travel  906f  miles? 

45.  A  body  falls  16^2"  ^^^^  *^®  ^^*^*  second,  and  in  each 
succeeding  second  321  feet  more  than  in  the  next  preced- 
ing.    How  far  does  it  fall  in  5  seconds  ? 

46.  Three  men.  A,  B,  and  C,  can  walk  around  a  circular 
race-course  in  8i,  7^,  and  6y\  minutes,  respectively.  If  they 
all  set  out  together,  after  how  many  minutes  will  they  all 
meet  at  the  starting-point,  and  how  many  times  will  each 
have  gone  around  the  course  ? 


FRACTIONS.  73 

47.  A  leaves  Boston  at  a  certain  time,  and  travels  at  the 
rate  of  3^  miles  an  hour.  After  he  has  been  gone  2|-  hours, 
B  sets  out  to  overtake  him,  and  travels  at  the  rate  of  4| 
miles  an  hour.  How  far  apart  are  A  and  B  5f  hours  after 
B  sets  out? 

48.  A  can  reap  a  field  in  9  days,  working  8  hours  a  day ; 
B  can  reap  the  same  field  in  8  days,  working  7^  hours  a 
day.  How  long  will  it  take  both  together  to  reap  the  field, 
working  9  hours  a  day  ? 

-  49.  The  sides  of  a  field  are  23|  rods,  60|  rods,  23^  rods, 
and  58|-  rods,  respectively.  What  is  the  length  of  the  long- 
est pole  that  will  be  contained  exactly  in  each  side  ? 

^    50.  Multiply  ^  of  f  of  9i  by  one-half  of  itself,  and  divide 
the  product  by  -^. 

51.  A  can  do  a  piece  of  work  in  9  hours  ;  A  and  B  to- 
gether can  do  it  in  6  hours,  and  B  and  C  together  can  do  it 
in  4  hours.  How  many  hours  will  it  take  A  and  C  together 
to  do  the  work? 

■--  52.  What  number  is  that  i%  of  |-|  of  which  exceeds  f  of 
i^of  itby  2|4? 

-'  53.  A  pole  stands  ^  in  the  mud,  -^  in  the  water,  and  the 
remainder,  12J  feet,  above  water.  Find  the  length  of  the 
pole. 

-  54.  A  sum  of  money  was  divided  between  A,  B,  C,  and 
D,  in  such  a  way  that  A  received  -f^,  B  -f-^,  C  -^j,  and  D  the 
remainder,  which  was  f  Q5^.  What  was  the  sum  divided, 
and  how  much  did  each  receive  ? 

""^  55.  If  17  horses  consume  S\  bushels  of  oats  in  3J  days, 
how  many  bushels  will  12  horses  consume  in  6f  days  ? 

"  56.  A  can  do  a  piece  of  work  in  12  days,  B  in  14  days,  C 
in  18  days,  and  D  in  21  days.  How  long  will  it  take  all  of 
them  together  to  do  the  work,  and  what  part  of  the  work 
does  each  perform  ? 


74  ARITHMETIC. 


X.    t)EOIMALS. 

113.  A  fraction  whose  denominator  is  a  power  of  ten  is 
usually  expressed  by  placing  a  point  at  the  right  of  the 
numerator,  and  then  moving  it  to  the  left  as  many  places  as 
there  are  ciphers  in  the  denominator. 

When  thus  expressed,  the  fraction  is  called  a  Decimal 
Fraction,  or  simply  a  Decimal. 

The  point  is  called  a  Decimal  Point. 

114.  Consider,  for  example,  the  fraction  f§^|-. 

In  this  case  there  are  three  ciphers  in  the  denominator. 
Placing  a  point  at  the  right  of  the  2305,  and  then  moving 
it  three  places  to  the  left,  we  have 

HM  =  2.305 

If  the  number  of  digits  in  the  numerator  is  less  than  the 
number  of  ciphers  in  the  denominator,  ciphers  may  be 
written  in  the  places  to  the  left  of  the  first  digit  of  the 
numerator. 

Thus,  consider  the  fraction  y^VV?r- 

Placing  a  point  at  the  right  of  the  16,  moving  it  four 
places  to  the  left,  and  writing  two  ciphers  at  the  left  of  the 
first  digit,  we  have 

■n?^  =  .0016 

115.  The  figure  immediately  to  the  right  of  the  decimal 
point  is  said  to  be  in  the  first  decimal  place;  the  next  one 
to  the  right  in  the  second  decimal  place  ;  etc. 

The  following  table  gives  the  signification  of  each  of  the 
first  six  decimal  places  : 

1st  ;  tenths.  4th ;  ten-thousandths. 

2d  ;  hundredths.  5th ;  hundred-thousandths. 

3d  J  thousandths.  6th  j  millionths. 


DECIMALS.  76 

116.  To  read  a  decimal,  first  read  the  number  to  the  left  of 
the  decimal  point,  if  any ;  then  the  number  to  the  right 
of  the  point,  regarded  as  an  integer,  followed  by  the  name 
of  the  right-hand  decimal  place. 

Thus,  2,305  is  read  "two,  and  three  hundred  and  five 
thousandths." 

.0016  is  read  "sixteen  ten-thousandths." 

In  order  to  avoid  ambiguity,  it  is  better  to  make  a  pause 
at  the  decimal  point,  and  another  before  pronouncing  the 
name  of  the  right-hand  decimal  place. 

EXAMPLES. 

117.  Kead  the  following : 

1.  .5.  6.  .0039.  11.  257400009. 

2.  .17.  7.  8.028.  12.  .0004859. 

3.  15.3.  8.  24.0071.  13.  863.108642. 

4.  .461„  9.  .84072.  14.  5.9085495. 

5.  90.06.  10.  .689313.  16.  .00003287. 

Write  the  following  as  decimals : 

16.  Forty-nine  hundredths. 

17.  Fifty-two,  and  four  tenths. 

18.  One  hundred  and  fifty-eight  thousandths. 

19.  Nine,  and  thirteen  ten-thousandths. 

20.  Thirty-seven,  and  two  hundred-thousandths. 

21.  Fifty-nine  thousand  three  hundred  and  ninety-eight 
millionths. 

22.  Eight  hundred  and  thirty-two,  and  forty  thousand 
one  hundred  and  two  hundred-thousandths. 

23.  Twenty-six,  and  eight  hundred  and  five  thousand 
three  hundred  and  three  millionths. 

24.  Seven  thousand  four  hundred  and  twenty-five  ten- 
millionths. 


76  ARITHMETIC. 

TO  REDUCE  A  DECIMAL  TO  A  COMMON  FRACTION. 

118.  A  decimal  may  be  expressed  in  the  form  of  a  com- 
mon fraction  by  writing  the  decimal  without  its  decimal 
point  for  a  numerator,  and  for  a  denominator  1,  followed  by 
as  many  ciphers  as  there  are  places  to  the  right  of  the 
decimal  point. 

Thus,  11.28  =  iji^=.^/; 

•0523  =  ^f|fo;  etc. 

EXAMPLES. 
Express  as  common  fractions  in  their  lowest  terms : 


1. 

2.8. 

6. 

8.512. 

11. 

.0376. 

16. 

.01375. 

2. 

.005. 

7. 

30.75. 

12. 

.0096. 

17. 

4.4375. 

3. 

75.44. 

8. 

.1975. 

13. 

3.0875. 

18. 

.15625. 

4. 

.684. 

9. 

68.461. 

14. 

.08309. 

19. 

.008128. 

6. 

1.85. 

10. 

.025. 

16. 

.00128. 

20. 

2.109375. 

ADDITION  OF  DECIMALS. 

119.  1.   Add  7.89,  31.4,  and  .086. 

7.89  We  write  the  numbers  so  that  their  decimal  points 

31.4  shall  be  in  the  same  vertical  column. 

QgQ  The  sum  of  8  hundredths  and  9  hundredths  is  17 

hundredths,  or  1  tenth  and  7  hundredths. 

39.376,  Ans.  The  sum  of  1  tenth,  4  tenths,  and  8  tenths  is  13 

tenths,  or  1  unit  and  3  tenths. 
The  sum  of  1  unit,  1  unit,  and  7  units,  is  9  units. 
Then  the  required  result  is  3  tens,  9  units,  3  tenths,  7  hundredths, 
and  6  thousandths,  or  39.376. 

EXAMPLES. 
Add  the  following : 

2.  25.5,  .00076,  1.7862,  and  .084. 

3.  2.601,  .9693,  35.08,  and  .00745. 


DECIMALS.  77 

4.  165,  .94468,  .0051,  and  59.226. 

5.  .4085,  8.62,  .03947,  and  2.139. 

6.  5.0902,  .00007,  .637,  and  .014961. 

7.  .39665,  9.9,  72.1508,  and  .004052. 

8.  .000616,  93.38967,  .0562,  and  807.74. 

9.  .06212,  35.49,  87.56,  4920.04,  and  297.868. 

SUBTRACTION    OF    DECIMALS. 
120.  1.  Subtract  89.725  from  162.0738. 
162.0738  We  write  the  numbers  so  that  their  decimal 

89  725  points  shall  be  in  the  same  vertical  column, 

■"ZT^rTTT      .  5  thousandths  from  13  thousandths  leave  8  thou- 

72.3488,  ^ns.     ^^„^^^^, 

3  hundredths  from  7  hundredths  leave  4  hundredths. 
7  tenths  from  10  tenths  leave  3  tenths. 
10  units  from  12  units  leave  2  units. 
9  tens  from  16  tens  leave  7  tens. 

Then  the  required  result  is  7  tens,  2  units,  3  tenths,  4  hundredths, 
8  thousandths,  and  8  ten-thousandths,  or  72.3488. 

If  the  subtrahend  has  more  places  than  the  minuend,  we 
may  make  the  number  of  places  in  the  latter  the  same  as 
in  the  former  by  mentally  supplying  ciphers  in  the  missing 
places. 

2.  Subtract  .008504  from  .0162. 
.0162 
.008504 
.007696,  Ans. 

EXAMPLES. 
Subtract  the  following : 

3.  .4169  from  5.2705.  8.  .005341  from  .0091291. 

4.  .0726  from  .32933.  9.  .0623907  from  10. 

5.  .318  from  1.  10.  .08194812  from  2.52866. 

6.  .00986  from  .0204.  11.  48.6007  from  830.352. 

7.  2.08429  from  11.352.  12.  .0002584  from  .0T683. 


78  ARITHMETIC. 

MULTIPLICATION  OF  DECIMALS. 

121.   1.   Multiply  30.84  by  2.516. 

Writing  the  decimals  as  common  fractions,  we  have 

30.84  X  2.516  =  ^^M  X  ?515  =  3084  x  2516. 
100       1000  100000 

3084 
2516 


18504 

3084 
15420 
6168 
7759344 

Then,  30.84  x  2.516  =  ^^j^^^-  =  77.59344,  Ans. 
It  is  customary  to  arrange  the  work  as  follows : 

30.84 

2.516 

18504 

3084 
15  420 
6168 
77.59344      ^ 

It  will  be  observed  that  the  number  of  decimal  places  in 
the  result  is  the  sum  of  the  number  of  decimal  places  in 
the  multiplicand  and  the  number  of  decimal  places  in  the 
multiplier ;  hence  the  following 

RULE. 

Multiply  the  numbers  as  if  they  were  integers,  and  point 
off  as  many  decimal  places  in  the  result  as  the  sum  of  the  num- 
ber of  decimal  places  in  the  multiplicand  and  multiplier. 

If  the  number  of  digits  in  the  product  is  not  sufficient  for 
this  purpose,  ciphers  may  be  written  in  the  places  to  the 
left  of  its  first  digit. 


DECIMALS.  79 

2.   Multiply  .764  by  .0108. 

.764 

.0108  In  this  case,  we  point  off  seven  decimal  places 

6112  in  the  product,  writing  two  ciphers  at  the  left  of 

764  the  first  digit. 


.0082512,  Ans. 

EXAMPLES. 
Multiply  the  following : 

3.  8.27  by  29.3.  10.  .5114  by  .4053. 

4.  .0966  by  .561.  11.  .068022  by  .16. 

5.  .00708  by  .0365.  12.  .4486  by  5.83. 

6.  .6581  by  9.7.  13.  18.052  by  .75. 

7.  .05648  by  .082.  14.   21.96  by  4.78. 

8.  1.821  by  34.5.  15.   .07819  by  63.05. 

9.  45.66  by  .00207.  16.    .009256  by  .08219. 

122.  To  Multiply  a  Decimal  by  10,  100, 1000,  Etc. 

To  multiply  a  decimal  by  10,  100,  etc.,  we  move  its  deci- 
mal point  one,  twd,  etc.,  places  to  the  right. 

Or  in  general,  to  multiply  a  decimal  by  1  followed  by  any 
number  of  ciphers,  we  move  its  decimal  point  to  the  right 
as  many  places  as  there  are  ciphers  in  the  multiplier. 

Example.   Multiply  87.35  by  10000. 

Moving  the  decimal  point /owr  places  to  the  right,  we  have 

87.35  X  10000  =  873500,  Ans. 

123.  To  Multiply  a  Decimal  by  .1,  .01,  .001,  Etc. 

To  multiply  a  decimal  by  .1,  .01,  .001,  etc.,  we  move  its 
decimal  point  one,  two,  three,  etc.,  places  to  the  leji. 

Or  in  general,  to  multiply  a  decimal  by  .1,  or  by  1  pre- 
ceded by  any  number  of  ciphers  and  then  a  decimal  point, 
we  move  its  decimal  point  as  many  places  to  the  left  as 
there  are  places  in  the  multiplier. 


80  ARITHMETIC. 

Example.   Multiply  6.294  by  .001. 

Moving  the  decimal  point  three  places  to  the  left,  we  have  .  ^ 

6.294  X  .001  =  .006294,  Ans. 

124.  To  Multiply  a  Decimal  by  Any  Number  of  Tens, 
Hundreds,  Etc. 

Any  number  of  ciphers  at  the  right  of  the  multiplier 
may  be  omitted,  if  the  decimal  point  of  the  multiplicand 
be  moved  to  the  right  as  many  places  as  there  are  ciphers 
omitted. 

Example.   Multiply  32.851  by  5200. 

3285.1 
52  We  move  the  decimal  point  of  32.851   two 


6570  2  places  to  the  right,  and  multiply  3285.1  by  52. 

164255  The  result  is  170825.2. 

170825.2,  Ans. 

In  like  manner,  ciphers  at  the  right  of  the  multiplicand 
may  be  omitted,  if  the  decimal  point  of  the  multiplier  be 
moved  to  the  right  as  many  places  as  .there  are  ciphers 
omitted. 

EXAMPLES. 
125.   Multiply  the  following  : 

1.  85.2  by  10.  9.  5839  by  .0001. 

2.  377  by  .01.  10.  1.417  by  10000. 

3.  4.14  by  300.  11.  368.8  by  .00001. 

4.  .00695  by  1000.  12.  .04854  by  89000. 

5.  .000208  by  100.  13.  937620  by  .001. 

6.  .753  by  1620.  14.  .060239  by  100000. 

7.  .1261  by  .1.  15.  14537000  by  .0985. 

8.  87900  by  .0743.  16.  27.405  by  5240000. 


DECIMALS.  81 

DIVISION    OF  DECIMALS. 

126.   Example.   Divide  4742.66  by  754. 

Writing  the  decimal  as  a  common  fraction, 

4742.66  --  754  =  ^^\^-^  -f-  754. 

754)474266(629 
4524 
2186 
1508 
6786 
6786 

Then,  4  7^y^6  6  _^  754  =  fj 9  ^  5.29,  Ans. 

It  is  customary  to  arrange  the  work  as  follows : 

754)4742.66(6.29 
4524 
218  6 
150  8 


6786 
67  86 


It  will  be  observed  that  the  number  of  decimal  places  in 
the  quotient  is  the  same  as  the  number  of  decimal  places  in 
the  dividend. 

127.  If  the  divisor  is  not  an  integer,  it  may  always  be 
made  so  by  moving  the  decimal  points  of  both  dividend' and 
divisor  as  many  places  to  the  right  as  there  are  decimal  places 
in  the  divisor. 

1.  Divide  .0275918  by  .7261. 

(.038,  Ans. 

7261)275.918  ^^  *^^^  ^^^^'  ^^  move  the  decimal  points 

217  83  ^^  ^*-**^  dividend  and  divisor  four  places 

ro  ACQ  to  the  right,  and  point  of£  three  places  in 

fjQ  Qoo  the  quotient. 


82  l^RITHMETIC.  t 

It  is  convenient,  in  Long  Division  of  Decimals,  to  write 
the  quotient  above  the  dividend  in  such  a  way  that  each  of 
its  digits  shall  be  directly  over  the  right-hand  digit  of  the 
corresponding  partial  product. 

Thus,  in  Ex.  1,  the  digit  3  of  the  quotient  is  directly 
over  the  right-hand  digit  of  the  first  partial  product,  21783, 
and  the  digit  8  is  directly  over  the  right-hand  digit  of  the 
second  partial  product,  58088. 

In  this  case,  the  decimal  point  of  the  quotient  will  always 
be  directly  over  the  decimal  point  of  the  dividend. 

If  the  number  of  decimal  places  in  the  dividend  is  less 
than  the  number  of  decimal  places  in  the  divisor,  ciphers 
may  be  written  in  the  missing  places. 

2.  Divide  318.68  by  .257. 

(1240,  Ans. 
257)318680 

^^^  In  this  case,  we  move  the  decimal  points  of 

616  both  dividend  and  divisor  three  places  to  the 

514  right,  annexing  one  cipher  to  the  dividend. 

1028 
1028 


If  the  dividend  is  not  exactly  divisible  by  the  divisor,  it 
may  sometimes  be  made  so  by  annexing  ciphers. 

3..  Divide  211.347  by  40.84. 

(5.175,  Ans. 
4084)21134.700 
20420 

714  7 

AQo  *  In  this  case,  we  annex  two  ciphers  to  the 

o^^  o/^  dividend  to  make  it  divisible  by  the  divisor. 

oOb  oU 

285  88 


20  420 
20  420 


DECIMALS.  83 

EXAMPLES. 
Divide  the  following : 

4.  4.361  by  .7.  14.  .0201474  by  .054. 

5.  .11504  by  .0004.  15.  .00113291  by  .193. 

6.  .005088  by  .06.  16.  19.7635  by  8.41. 

7.  .9588  by  9.4.  17.  .31236  by  .00685. 

8.  284.24  by  .038.  18.  46.290881  by  .9107. 

9.  19.3752  by  20.7.  19.  .487578  by  .00665. 

10.  4.6292  by  2.84.  20.  6.618015  by  7.174. 

11.  .492453  by  .549.  21.  609.4429  by  .001243. 

12.  .313048  by  87.2.  22.  332.45  by  .0488. 

13.  29379.7  by  .47.  23.  .35363808  by  89.28. 

128.  To  Divide  a  Decimal  by  10,  100,  1000,  Etc. 

To  divide  a  decimal  by  10,  100,  etc.,  we  move  its  decimal 
point  one,  two,  etc.,  places  to  the  left. 

Or  in  general,  to  divide  a  decimal  by  1  followed  by  any 
number  of  ciphers,  we  move  its  decimal  point  to  the  left 
as  many  places  as  there  are  ciphers  in  the  divisor. 

Example.   Divide  87.35  by  10000. 
Moving  the  decimal  point  four  places  to  the  left,  we  have 
87.35  --  10000  =  .008735,  Ans. 

129.  To  Divide  a  Decimal  by  .1,  .01,  .001,  Etc. 

To  divide  a  decimal  by  .1,  .01,  .001,  etc.,  we  move  its 
decimal  point  one,  two,  three,  etc.,  places  to  the  right. 

Or  in  general,  to  divide  a  decimal  by  .1,  or  by  1  preceded 
by  any  number  of  ciphers  and  then  a  decimal  point,  we  move 
its  decimal  point  as  many  places  to  the  right  as  there  are 
places  in  the  divisor. 

Example.    Divide  6.294  by  .0001. 
Moving  the  decimal  point  four  places  to  the  right,  we  have 
6.294  -f-  .0001  =  62940,  Ans. 


84  ARITHMETIC. 

130.  To  Divide  a  Decimal  by  Any  Number  of  Tens,  Hun- 
dreds, Etc. 

To  divide  a  decimal  by  any  number  of  tens,  hundreds, 
etc.,  we  omit  the  ciphers  at  the  right  of  the  divisor,  and 
move  the  decimal  point  of  the  dividend  as  many  places  to 
the  left  as  there  are  ciphers  omitted. 

Example.    Divide  4716.28  by  62800. 

(.0751,  Ans. 
628)47.1628 

^^  ^^  In  this  case,  we  move  the  decimal  point 

3  202  of  4716.28  two  places  to  the  left,  and  divide 

3140  the  result  by  628. 


628 
*628 

EXAMPLES 

>• 

131.   Divide  the 

following : 

1. 

542  by  100. 

8. 

.463  by  .0001. 

2. 

2.99  by  .01. 

9. 

7.815  by  1000. 

3. 

630  by  5000. 

10. 

.0008171  by  .001. 

4. 

.426  by  10. 

11. 

855.36  by  6480000. 

5. 

2697.5  by  8300 

12. 

5156.8  by  10000. 

6. 

.00724  by  .1. 

13. 

.001807  by  .00001. 

7. 

6.8337  by  270. 

14. 

30.82  by  100000. 

TO  REDUCE  A  COMMON  FRACTION  TO  A  DECIMAL. 
132.   1.   Reduce  -^^  to  a  decimal. 
We  have  -^-^  =  57  -=-  125 ;  dividing  57  by  125,  we  obtain 

(.456,  Ans. 
125)57.000 
50  0 
7  00 
6  25 
750 
750 


DE(tMi4ISl6yERS/TY    )  86 


2.   Eeduce  |  to  a  decimal, 

(.6666^,  Ans. 
3)2.0000 
18 
20 
18 
20 
18 
20 
18 
2 

In  this  case,  the  division  never  terminates,  no  matter  how 
far  the  operation  may  be  carried. 

Whenever  the  process  has  been  carried  as  far  as  desired, 
the  remainder  may  be  written  over  the  divisor,  and  the 
fraction  thus  formed  added  to  the  quotient. 

In  many  numerical  computations,  only  an  approximate 
value  of  the  quotient  is  required ;  in  such  a  case,  the  frac- 
tion which  expresses  the  remainder  may  be  omitted  provided 
that,  if  it  is  equal  to  or  greater  than  i,  the  last  digit  of  the 
quotient  is  increased  by  1. 

Thus,  .725^  would  be  taken  as  .725,  approximately ;  .62|- 
as  .63 ;  and  .6666|  as  .6667. 

The  result  .6667  is  said  to  be  the  approximate  value  of  j 
to  the  nearest  fourth  decimal  place,  or  to  the  nearest  ten-thou- 
sandth. 

Note.  It  may  sometimes  happen  that  neither  the  expression  of 
the  remainder,  nor  the  nearest  approximate  value  is  necessary ;  in 
such  a  case,  the  incompleteness  of  the  quotient  is  denoted  by  a  +  sign. 

Thus,  .6666+  would  be  written  in  place  of  Mm^. 

EXAMPLES. 
Reduce  the  following  to  decimals  : 


f 

6-  W- 

9-^- 

12.  tt- 

A- 

7.  AW 

10.  /j. 

13.  im- 

H- 

S-Tittrr- 

ll-rfir- 

14.   T3VW 

86  ARITHMETIC. 

Find  the  approximate  value  of  each,  of  the  following  to 
the  nearest  fifth  decimal  place : 

15.  f  17.  ^.  19.  ^^.  21.  Hf. 

16.  {i.  18.  ||.  20.  ^.  22.  HJ. 

CIRCULATING  DECIMALS. 

133.  In  expressing  a  common  fraction  as  a  decimal  by  the 
method  of  Art.  132,  if  the  factors  of  the  denominator  are 
not  all  2's  and  5's,  a  single  figure,  or  a  set  of  figures,  will 
be  found  to  recur  indefinitely  in  the  quotient. 

Thus,  if  17  be  divided  by  54,  the  quotient  is  .3148148+, 
where  the  digits  148  recur  indefinitely. 

Such  a  decimal  is  called  a  Circulating  Decimal,  or  simply 
a  Circulate ;  and  the  digit,  or  set  of  digits,  which  is  repeated 
is  called  the  Repetend. 

134.  A  circulate  is  usually  expressed,  when  a  single  digit 
is  repeated,  by  writing  a  dot  over  it;  and  when  a  set  of  digits 
is  repeated,  by  writing  dots  over  the  first  and  last  of  the  set. 

Thus,  .3148148  +  is  expressed  .3148. 

135.  A  Pure  Circulate  is  one  which  has  no  digits  except 
the  ones  which  are  repeated ;  as  .38. 

A  Mixed  Circulate  is  one  which  has  one  or  more  digits 
preceding  the  ones  which  are  repeated ;  as  4.3157. 

136.  To  Express  a  Common  Fraction  as  a  Circulate. 
1.  Express  -^J  as  a  circulate. 

(.3i48,  Ans. 

^ao  Dividing  17  by  54,  the  first  four  digits  of 

— — -  the  quotient  are  .3148. 

Z/\  At  this  point  a  remainder,  80,  is  obtained 

which  is  the  same  as  the  first  remainder. 

260  It  is  evident  from  this  that  the  digits  148 

^1^  will  recur  indefinitely  in  the  quotient. 

440  Then  the  required  result  is  .3148. 
432 

80 


DECIMALS.  87 

In  any  case,  the  division  must  be  carried  ont  until  a 
remainder  is  obtained  which  is  the  same  as  the  dividend^  or 
some  preceding  remainder. 

EXAMPLES. 

Express  each  of  the  following  as  a  circulate : 

2.  A-  5.  ^.  8.  3-V\V  11.  tA^. 

3.  3|.  6.  m.  9.  m.  12.  iUi. 

4.  if  7.  Idj^^.  10.  «i.  13.  7^\. 

137.  To  Find  the  Common  Fraction  which  will  produce  a 
Given  Circulate. 

1.  What  fraction  will  produce  .53  ? 

Let  F  represent  the  fraction. 
Then,  one-hundred  times  F  is  equal  to  53.53. 
Therefore,  one-hundred  times  F,  minus  F,  is  equal  to 
53.53,  minus  .53,  or  53. 

That  is,  ninety-nine  times  F  is  equal  to  53. 
Whence,  F  is  equal  to  53  divided  by  99,  or  -Jf ,  Ans. 

From  the  above  example,  we  derive  the  following 

RULE. 

To  find  the  common  fraction  which  will  produce  a  given 
pure  circulate,  divide  the  repetend  by  a  number  having  for  its 
digits  as  many  nines  as  there  are  digits  in  the  repetend. 

Thus,  the  fraction  which  will  produce  .353  is  ||^. 

2.  What  fraction  will  produce  .5185  ? 
By  the  above  rule,  the  fraction  is 

•^^-•^^^-10  "270"  27'^ 


3. 

.39. 

8. 

.327. 

4. 

.47. 

9. 

.0945. 

5. 

.8i. 

10. 

48.573. 

6. 

.407. 

11. 

.eosi. 

7. 

2.675. 

12. 

.3726. 

88  ARITHMETIC. 

EXAMPLES. 

rind  the  common  fraction  which  will  produce  each  of  the 
following : 

13.  .5243.  18.  .27128. 

14.  .9482.  19.  .51378. 

15.  .2075.  20.  .24259. 

16.  7.8456.  21.  5.072i6. 

17.  .60405.  22.  .36138. 

TO  MULTIPLY  OR  DIVIDE  A  NUMBER  BY  AN  ALIQUOT 
PART  OF   10,   100,   1000,   ETC. 

138.  An  Aliquot  Part  of  a  number  is  a  number  that  is 
exactly  contained  in  it. 

Thus,  S^  is  an  aliquot  part  of  10,  for  it  is  contained  in  10 
three  times. 

139.  1.   Multiply  5.736  by  16|. 

Since  16|  is  one-sixth  of  100,  we  may  multiply  5.736  by  100,  and 
divide  the  result  by  6. 

Multiplying  5.736  by  100,  the  product  is  573.6  (Art.  122). 
Dividing  573.6  by  6,  the  quotient  is  95.6,  Ans. 

2.   Divide  1.056  by  125. 

Since  125  is  one-eighth  of  1000,  we  may  divide  1.056  by  1000,  and 
multiply  the  result  by  8. 

Dividing  1.056  by  1000,  the  quotient  is  .001056  (Art.  128). 
Multiplying  .001056  by  8,  the  product  is  .008448,  Ans. 

EXAMPLES. 
Multiply  the  following : 

3.  72  by  ^.  7.  2.46  by  166|. 

4.  .84  by  25.  8.  .00047  by  125. 
6.  .0393  by  6}.         9.  .976  by  6J. 

6.  .00525  by  33^.       10.  175.2  by  83^. 


DECIMALS.  89 

Divide  the  following : 

11.  35  by  2f  15.  413  by  250. 

12.  10.1  by  50.  16.  .98  by  12f 

13.  .76  by  333i.  17.  6524  by  8^. 

14.  257  by  16|.  18.  802.9  by  62f 


MISCELLANEOUS   EXAMPLES. 

140.   1.  Add  .38752,  12.893,  .008245,  and  5.0169. 

2.  Subtract  .000695367  from  .00518224. 

3.  Express  .03616  as  a^common  fraction,  and  reduce  tbe 
result  to  its  lowest  terms. 

4.  Multiply  5629070  by  .000001. 

5.  Express  i-|  and  ff  as  decimals,  and  find  their  sum. 

6.  Divide  1.0642  by  78.25. 

7.  Express  -jy|-|  as  a  circulating  decimal. 

8.  Express  .25625  as  a  common  fraction,  and  reduce  the 
result  to  its  lowest  terms. 

9.  Multiply  84.175  by  .0007302.' 

10.  Divide  .015271938  by  .4521. 

11.  Multiply  .0297  by  111|. 

12.  Eeduce  yf  g  to  a  decimal. 

13.  Multiply  .00935  by  668000. 

14.  Divide  473.1  by  .0166. 

15.  What  common  fraction  will  produce  .9702  ? 

16.  Divide  .0603712  by  .00000001. 

17.  Eind  the  approximate  value  of  ^-^  to  the  nearest  fifth 
decimal  place. 

18.  Divide  6^6.6  by  7450000. 

19.  Divide  85.29  by  3333f 

20.  Express  ^  as  a  circulating  decimal. 


90  ARITHMETIC. 

21.  Express  ^^^  and  -^^-^  as  decimals,  and  subtract  the 
second  result  from  the  first. 

22.  Divide  .48868466  by  .005407. 

23.  What  common  fraction  will  produce  .51296  ? 


24.    Simplify -QQ^^  +  'Q^i 


2f-1.7 


25.  Simplify -^^-('^Q^  +  'QQ^^Q^^). 

•^       .9 -(.12 -.063) 

26.  Simplify  ^  X  -^  X  :^535. 

^     "^  .049      .0045       .96 

27.  Simplify  ^f:5^-i??Y 

^     -^  21.5V. 034       -85; 

28.  Simplify  :^^^^  +  :508+.5004. 

^     -^    .5  -  .025       .03  +  .0015 

29    ^ir^pi^fy  (-38  X  .00027)+ (.057  x  .0036) 
^     *^  1-.9487 

30.    Simplify  li+iM-iI^^H^. 
.16  +  .8      .71  -  .436 


UNITED  STATES  MONEY. 

141.  Money  is  that  which  is  used  to  measure  value. 

142.  A    Denomination    is    a   unit  of  measure  ;    as   for 
example,  a  dollar,  or  a  cent. 

The  denominations  of  United  States  Money  are  given  in 
the  following 

TABLE. 

10  mills  (m.)  =  1  cent.  (c.) 

10  cents  =  1  dime,  (d.) 

10  dimes  =  1  dollar.  ($) 

10  dollars        =  1  eagle,  (e.) 


DECIMALS.  91 

143.  The  only  denomijiations  used  in  ordinary  business 
transactions  are  dollars  and  cents;  eagles  and  dimes  are 
usually  expressed  in  terms  of  dollars  and  cents,  respectively, 
and  mills  as  the  fraction  of  a  cent. 

Thus,  5  eagles,  3  dollars,  7  dimes,  9  cents,  and  2  mills,  is 
the  same  as  53  dollars,  and  79|-  cents. 

Since  a  cent  is  one-hundredth  of  a  dollar,  any  sum  of 
money  expressed  in  dollars  and  cents  may  be  expressed  as  a 
decimal  of  a  dollar  by  writing  the  number  of  cents  in  the 
hundredths'  place. 

Thus,  53  dollars  and  79^  cents  may  be  expressed  as  53.79^ 
dollars,  or  $  53.79^. 

144.  A  sum  of  money  expressed  as  a  decimal  of  a  dollar 
may  be  expressed  as  a  decimal  of  a  dime  by  multiplying  by 
10 ;  as  a  decimal  of  a  cent  by  multiplying  by  100 ;  and  as  a 
decimal  of  a  mill  by  multiplying  by  1000. 

Hence  (Art.  122),  a  sum  of  money  expressed  as  a  decimal 
of  a  dollar  may  be  expressed  as  a  decimal  of  a  dime  by  mov- 
ing its  decimal  point  one  place  to  the  right ;  as  a  decimal  of 
a  cent  by  moving  the  decimal  point  two  places  to  the  right ; 
and  as  a  decimal  of  a  mill  by  moving  the  decimal  point 
three  places  to  the  right. 

Thus,  f  7.32  =  73.2  dimes  =  732  cents  =  7320  mills. 

145.  A  sum  of  money  expressed  as  a  decimal  of  a  mill 
may  be  expressed  as  a  decimal  of  a  cent  by  moving  its 
decimal  point  one  place  to  the  left ;  as  a  decimal  of  a  dime 
by  moving  the  decimal  point  two  places  to  the  left ;  and  as 
a  decimal  of  a  dollar  by  moving  the  decimal  point  three 
places  to  the  left. 

Thus,  7852.1  mills  =  785.21  cents  =  78.521  dimes  = 
$7.8521. 

146.  If  two  sums  of  money  be  expressed  as  decimals  of 
the  same  denomination,  they  may  be  added  or  subtracted  by 
the  methods  of  Arts.  119  or  120 ;  the  result  being  a  decimal 
of  the  same  denomination. 


92  ARITHMETIC. 

They  may  also  be  divided  by  the  method  of  Art.  127. 

Again,  a  sum  of  money  expressed  as  a  decimal  of  any 
denomination  may  be  multiplied  or  divided  by  the  methods 
of  Arts.  121  or  127  j  the  result  being  a  decimal  of  the  same 
denomination. 

1.  Add  together  $43.29,  1106.7  cents,  7480  mills,  and 
615  dimes. 

Reducing  each  sum  to  the  decimal  of  a  dollar,  by  the  rule  of  Art. 
145,  we  have, 

$43.29 
11.067 

7.48 
51.5 
$113,337,  Ans. 

2.  Divide  .0118364  dimes  by  .932  mills. 

Reducing  the  dimes  to  the  decimal  of  a  mill  by  moving  the  decimal 
point  two  places  to  the  right,  we  have 

.932)1.18364(1.27,  Ans. 
932 
2516 

1864 


6524 
6524 


EXAMPLES. 

3.  Express  $  19.29  as  a  decimal  of  a  mill. 

4.  Express  .000382  eagles  as  a  decimal  of  a  cent. 

5.  Express  453.7  dimes  as  a  decimal  of  an  eagle. 

6.  Express  29  mills  as  a  decimal  of  a  dollar. 

7.  Find  the  sum  of  $  115.28,  6325.2  cents,  8.3  dimes, 
and  47101  mills. 

8.  Find  the  sum  of  .437  dimes,  1055.4  mills,  $7.2195, 
and  543.96  cents. 


DECIMALS.  98 

9.    Subtract  3845  mills  from  $9.63. 

10.  Subtract  115.28  cents  from  39.07  dimes. 

11.  Subtract  $59,223  from  20091  cents. 

12.  Subtract  $  .092183  from  489.56  mills,  and  express  the 
result  as  a  decimal  of  an  eagle. 

13.  Multiply  $320.16  by  100. 

14.  Multiply  $  95.78  by  .0001. 

15.  Divide  $187.25  by  10000. 

16.  Divide  $42.56  by  .001. 

17.  Multiply  $73.29  by  380. 

18.  Multiply  $53.08  by  72.9. 

19.  Multiply  $  216.273  by  .414. 

20.  Multiply  302.8  mills  by  967,  and  express  the  result 
as  a  decimal  of  a  dime. 

21.  Divide  $308,238  by  86.1. 

22.  Divide  $210.60  by  5400. 

23.  Divide  $  669.90  by  2175  cents. 

24.  Divide  .410652  mills  by  .0561  cents. 

25.  Divide  15.2513  dimes  by  $  .349. 

26.  Divide  23.39888  cents  by  34.01  dimes. 

PROBLEMS. 

147.   1.   What  is  the  cost  of  27  pounds  of  tea  at  43^  cents 
a  pound  ? 

27 

g1  We  multiply  27  first  by  3,  then  by  4,  and  finally 

-jl  Q  g  by  i,  and  add  the  results. 

9 

$11.70,  ^ns. 


94  ARITHMETIC. 

2.  How  many  yards  of  cloth  at  66^  cents  a  yard  can  be 
bought  for  ^8.00? 

800  _  800  _  ^^^      _3_  $ 8.00  is  the  same  as  800  cents. 

66f  ""  ^^  ~"  ^^^      ^00  Dividing  800  cents  by  66 1  cents,  the 

=  12,  Ans.        quotient  is  12. 

3.  What  is  the  cost  of  3|  tons  of  coal  at  $  5.76  a  ton  ? 

4.  What  is  the  cost  of  147  yards  of  cloth  at  16f  cents  a 
yard? 

5.  Find  the  sum  of  |i  of  $  5.67,  and  f  of  $  8.75. 

6.  How  many  pounds  of  coffee,  at  41|  cents  a  pound,  can 
be  bought  for  ^121.25? 

7.  If  35352  yards  of  silk  cost  $  59.78,  how  much  will  one 
yard  cost  ? 

8.  What  is  the  cost  of  18  tons  of  coal  at  $5.83J  a  ton  ? 

9.  A  grocer  received  on  Monday  $135.25,  on  Tuesday 
$84.40,  on  Wednesday  $106.65,  on  Thursday  $122.70,  on 
Friday  $  93.62,  and  on  Saturday  $185.56.  What  were  his 
total  receipts  for  the  week  ? 

10.  What  is  the  cost  of  a  barrel  of  sugar  weighing  276 
pounds,  at  $  .04f  a  pound  ? 

11.  What  is  the  cost  of  17|  acres  of  land  at  $  238.45  an 
acre? 

12.  If  12|  cords  of  wood  cost  $116.66,  how  much  will 
one  cord  cost  ? 

13.  If  bricks  be  sold  at  $  6.75  a  thousand,  how  much  will 
8960  bricks  cost  ? 

14.  How  many  barrels,  each  containing  32.75  gallons,  can 
be  filled  from  1200  gallons  of  wine,  and  how  many  gallons 
will  remain  ? 

15.  How  many  barrels  of  flour,  at  $  7.35|  a  barrel,  can  be 
bought  for  $551.87^? 

16.  If  13f  yards  of  cloth  can  be  bought  for  $33.20,  how 
much  will  15|  yards  cost  ? 


DECIMALS.  95 

17.  If  24  pounds  of  butter  be  given  for  100.8  pounds  of 
sugar,  bow  many  pounds  sbould  be  given  for  91  pounds  of 
sugar  ? 

18.  A  man  having  ^100,  spent  $28.59,  then  received 
$15.75,  then  spent  $48.98,  and  finally  received  $37.37. 
How  much  money  then  had  he  ? 

19.  A  farmer  sold  15  loads  of  wheat,  each  containing  8.75 
bushels,  for  92  cents  a  bushel.  How  much  money  did  he 
receive  ? 

20.  If  51  tons  of  coal  cost  $  50,  how  much  will  fl-  of  a 
ton  cost  ? 

21.  The  product  of  three  numbers  is  3.9701556.  If  two 
of  them  are  92.7  and  5.16,  what  is  the  third  ? 

22.  If  13|  barrels  of  flour  cost  $  89.11,  how  much  will  || 
of  a  barrel  cost  ? 

23.  A,  B,  and  C  received  $17.40  for  a  piece  of  work. 
If  A  did  -f^  of  the  work,  B  -^-^,  and  C  the  remainder,  how 
much  money  should  each  receive  ? 

24.  A  man  worked  25f  days.  He  paid  out  |  of  his  earn- 
ings for  board,  and  had  $  12.40  left.     Find  his  daily  wages. 

25.  A  and  B  start  at  the  same  time,  from  the  same  place, 
and  walk  in  the  same  direction  at  the  rates  of  3.4798  and 
4.1263  miles  an  hour,  respectively.  How  far  apart  are  they 
at  the  end  of  7.255  hours  ? 

26.  A  dealer  sold  goods  for  $124.25,  and  gained  |  of 
what  they  cost  him.     How  much  did  they  cost  him  ? 

27.  Which  is  the  greater,  -J  of  $16.12,  or  |  of  $17.67, 
and  how  much  ? 

28.  Find  the  cost  of  18|  feet  of  steel  rod,  at  $  .35  a  foot, 
and  241  feet  at  $  .456  a  foot. 

29.  A  merchant  sold  goods  for  $  87.95,  and  lost  f  of  what 
they  cost  him.     How  much  did  he  lose  by  the  operation? 

30.  If  15f  yards  of  cloth  can  be  bought  for  $12.73,  how 
many  yards  can  be  bought  for  $  18.76  ? 


96  ARITHMETIC. 

31.  A  man  bought  7  car-loads  of  wheat,  each  car  con- 
taining 165  bushels,  at  ^  .84^  a  bushel.    What  was  the  cost  ? 

32.  If  5f  pounds  of  coffee  can  be  bought  for  $  2.36^,  how 
many  pounds  can  be  bought  for  $6.54^? 

33.  If  A  can  do  a  piece  of  work  in  2.4  days,  and  B  in  3.2 
days,  how  long  will  it  take  both  of  them  together  to  do  the 
work? 

34.  Find  the  cost  of  237  bales  of  silk,  each  containing 
125|  yards,  at  $1.26  a  yard. 

35.  A  man  spent  -f  of  his  money  for  provisions,  ^  of  the 
remainder  for  clothing,  and  had  f  12.87  left.  How  much  had 
he  at  first  ? 

36.  If  .488  of  a  ton  of  coal  be  worth  $  3.05,  how  much 
will  7.56  tons  cost? 

37.  A  man  left  ^  of  his  estate  to  his  wife,  f  of  the  re- 
mainder to  his  son,  and  the  rest  to  his  daughter.  The  wife 
received  $  546.75  more  than  the  daughter.  What  did  each 
receive  ? 

38.  A  farmer  sold  35  tubs  of  butter,  each  weighing  43^ 
pounds,  at  21|-  cents  a  pound.  He  bought  21  barrels  of 
flour  at  $6f  a  barrel,  and  received  the  balance  in  cash. 
How  much  money  did  he  receive  ? 

39.  Three  men.  A,  B,  and  C,  can  do  a  piece  of  work  in 
18,  24,  and  36  hours,  respectively.  How  long  will  it  take 
all  of  them  together  to  do  the  work?  If  they  receive 
$  11.25  for  the  work,  how  should  the  money  be  divided  ? 

40.  A  merchant  bought  goods  to  the  amount  of  $  228.60. 
He  kept  y^-g-  of  them  for  his  own  use,  and  sold  the  remainder 
for  -^  more  than  they  cost  him.     How  much  did  he  gain  ? 

41.  A  grocer  bought  10  pounds  of  tea  at  $  .38^  a  pound, 
12  pounds  at  $.41|-  a  pound,  and  15  pounds  at  $.433  a 
pound.  He  sold  the  whole  at  $  .44^  a  pound.  How  much 
did  he  gain  ? 


DECIMALS.  97 

42.  A  gentleman  divided  f  22.05  between  his  two  sons  in 
such  a  way  that  the  younger  received  f  as  much  as  the 
elder.     How  much  did  each  receive  ? 

43.  If  a  horse  travels  8.3  hours  a  day,  at  the  rate  of  6.25 
miles  an  hour,  how  many  days  will  it  take  him  to  travel 
830  miles  ? 

44.  Find  the  cost  of  12.6  bales  of  silk,  each  containing 
73.625  yards,  at  $  1.20  a  yard. 

45.  A  tank  has  two  pipes.  One  of  them  can  fill  it  in 
8.5  minutes,  and  the  other  can  empty  it  in  12.75  minutes. 
How  many  minutes  will  it  take  to  fill  the  tank,  if  both 
pipes  are  opened  together  ? 

46.  A  man  invests  f  of  his  property  in  real  estate,  -^ 
of  the  remainder  in  railway  shares,  and  the  balance  in 
city  bonds.  The  amount  invested  in  city  bonds  exceeds 
by  $93.75  the  amount  invested  in  real  estate.  Find  the 
amount  of  each  kind  of  investment. 

47.  A  gentleman  left  ^  of  his  property  to  his  wife,  ^  to 
his  elder  son,  i  to  his  younger  son,  i  to  his  daughter,  and 
the  balance,  $369.75,  to  a  charitable  institution.  How 
much  did  each  receive  ? 

48.  A,  B,  and  C  can  do  a  piece  of  work  in  6,  14,  and  21 
days,  respectively.  B  and  C  worked  alone  for  5  days,  when 
they  were  joined  by  A,  and  the  work  was  completed  by  all 
of  them  together.  If  they  received  $  54  for  the  work,  how 
should  the  money  be  divided  ? 

49.  Three  pipes  can  empty  a  tank  in  2.25,  3.375,  and 
5.0625  minutes,  respectively.  How  many  minutes  will  it 
take  to  empty  the  tank  if  all  the  pipes  are  opened  ? 


98  ARITHMETIC. 

XI.   MEASURES. 

148.  Measures  of  Length. 

Measures  of  Length,  or  Linear  Measures,  are  those  used  in 
measuring  lengths  or  distances. 

TABLE. 

12    inches  (in.)  =1  foot,  (ft.) 

3    feet  =lyard.  (yd.) 

5 J  yards  =  1  rod.  (rd.) 

320    rods  =  1  mile,  (mi.) 

It  follows  from  the  above  that 

1  mile  =  1760  yards  =  5280  feet. 

Surveyors  use,  in  the  measurement  of  land,  a  chain  (ch.) 
whose  length  is  4  rods,  divided  into  100  links  (li.)  of  7.92 
inches  each. 

80  chains  are  equal  to  one  mile. 

149.  Measures  of  Area. 

Measures  of  Area,  Surface  Measures,  or  Square  Measures, 

are  those  used  in  measuring  areas. 

TABLE. 

144    square  inches  (sq.  in.)  =  1  square  foot.  (sq.  ft.) 

9    square  feet  =  1  square  yard.  (sq.  yd.) 

30^  square  yards  =  1  square  rod.  (sq.  rd.) 

160    square  rods  =  1  acre.     (A.) 

640    acres  =  1  square  mile.  (sq.  mi.) 

It  follows  from  the  above  that 

1  acre  =  43560  square  feet. 
10  square  chains  are  equal  to  one  acre. 


MEASURES.  99 

150.  Measures  of  Volume. 

Measures  of  Volume,  or  Cubic  Measures,  are  those  used  in 
measuring  volumes. 

TABLE. 

1728  cubic  inches  (cu.  in.)  ==  1  cubic  fpot.     (cu.  ft.) 
27  cubic  feet  =  1  cubic  yard.    (cu.  yd.) 

The  following  is  used  in  measuring  wood : 
128  cubic  feet  =  1  cord,     (cd.) 

151.  Measures  of  Capacity. 

Liquid  Measures  are  used  in  measuring  liquids. 

TABLE. 

4  gills  (gi.)  =  l  pint.       (pt.) 
2  pints  =  1  quart,     (qt.) 

4  quarts        =  1  gallon,   (gal.) 

The  following  are  less  frequently  used : 

31J  gallons  =  1  barrel. 
63    gallons  =  1  hogshead. 

Dry  Measures  are  used  in  measuring  grain,  vegetables, 
fruit,  etc. 

TABLE. 

2  pints  (pt.)  =1  quart,  (qt.) 
8  quarts  =  1  peck.  (pk.) 
4  pecks  =1  bushel,    (bu.) 

The  quart  liquid  measure  contains  57f  cubic  inches,  and 
the  quart  dry  measure  67-|-  cubic  inches  ;  the  gallon  contains 
231  cubic  inches,  and  the  bushel  2150.42  cubic  inches. 


100  ARITHMETIC. 

Apothecaries'  Liquid  Measures  are  used  in  compounding 
medicines. 

TABLE. 

60  minims  (n\^)=l  fluid  dram.      (f3) 
8  fluid  drams    =  1  fluid  ounce,     (f  5  ) 
16  fluid  ounces  =  1  pint.     (0.) 

152.  Measures  of  Weight. 

Avoirdupois  Weight  is  used  in  weighing  all  common 
articles. 

TABLE. 

16  drams  (dr.)         =  1  ounce,     (oz.) 
16  ounces  =  1  pound,     (lb.) 

100  pounds  =  1  hundred-weight,     (cwt.) 

20  hundred-weight  =  1  ton.     (T.) 

The   following  are  used  at  the  United   States   Custom 
Houses,  and  in  weighing  iron  and  coal  at  the  mines : 

112  pounds  =  1  long  hundred-weight. 
2240  pounds  =  1  long  ton. 

Troy  Weight  is  used  in  weighing  gold,  silver,  and  jewels. 

TABLE. 
24  grains  (gr.)     =  1  pennyweight,     (pwt.) 
20  penny  weights  =  1  ounce,     (oz.) 
12  ounces  =  1  pound,     (lb.) 

Apothecaries'  Weight  is  used  in  compounding  medicines. 

TABLE. 

20  grains  (gr.)=l  scruple.     (3) 

3  scruples        =  1  dram.         (  3  ) 

8  drams  =  1  ounce.       ( 5 ) 

12  ounces  =  1  pound.       (tb) 


MEASURES.  101 

A  pound  avoirdupois  contains  7000  grains,  and  a  pound 
troy  5760  grains;  the  pound,  ounce,  and  grain,  in  apothe- 
caries' weight,  have  the  same  weight  as  in  troy  weight. 


153.  Measures  of  Time 

TABLE. 

60  seconds  (sec.) 

=  1  minute,     (min.) 

60  minutes 

=  1  hour. 

(h.) 

24  hours 

=  1  day. 

(d.) 

7  days 

=  1  week. 

(wk.) 

365  days 

=  1  common  year,     (y.) 

366  days 

=  1  leap  year. 

The  Calendar  Months  are  as  follows : 

January, 

31  dayj 

3. 

July, 

31 

iays 

February,  28  oi 

29     " 

August, 

31 

March, 

31     " 

September, 

30 

April, 

30    " 

October, 

31 

May, 

31     " 

November, 

30 

June, 

30     " 

December, 

31 

The  Solar  Year  is  365  days,  5  hours,  48  minutes,  and 
49.7  seconds,  or  very  nearly  365^  days. 

If  the  number  of  any  year  is  divisible  by  4,  the  month  of 
February  has  29  days,  and  the  year  is  called  a  leap  year; 
but  if  the  number  of  the  year  is  divisible  by  100,  it  is  not  a 
leap  year,  unless  it  is  divisible  by  400. 

Thus,  1600  is  a  leap  year,  but  not  1700. 

154.  English  Money. 

TABLE. 

4  farthings  (far.)  =  1  penny,     (d) 
12  pence  =  1  shilling,    (s.) 

20  shillings  =  1  pound.      (£) 


102  ARITHMETIC. 

The  following  are  also  used : 

1  florin  =  2  shillings. 
1  crown  =  5  shillings. 
1  guinea  =  21  shillings. 

The  English  coin  of  the  value  of  20  shillings  is  called  a 
sovereign. 

155.  Angular  Measures. 

Angular  Measures  are  used  in  measuring  angles  and  arcs 
of  circles. 

TABLE. 

60  seconds  (")  =  1  minute.     (') 
60  minutes        =  1  degree.      (°) 

The  following  are  also  used : 

A  quadrant  is  an  arc  of  90°. 

A  circumference  is  an  arc  of  360°. 

A  right-angle  is  an  angle  of  90°. 

The  length  of  a  degree  (-jj^)  of  the  earth's  equator  is 
about  69|-  miles. 

156.  Miscellaneous  Tables. 

Numbers.  Paper. 

12  units  =  1  dozen.  24  sheets    =  1  quire. 

12  dozen  =  1  gross.  20  quires    =  1  ream. 
12  gross  =  1  great  gross.  2  reams    =  1  bundle. 

20  units  =  1  score.  5  bundles  =  1  bale. 


DENOMINATE  NUMBERS.  103 


XII.    DENOMINATE    NUMBERS. 

157.  A  Denomination  is  a  unit  of  measure;  as  for  ex- 
ample, a  mile,  a  pound,  or  a  bushel.     (Compare  Art.  142. ) 

158.  Two  or  more  denominations  are  said  to  be  of  the 
same  kind  when  each  can  be  expressed  in  terms  of  the  others. 

Thus,  a  mile  and  a  rod  are  denominations  of  the  same 
kind. 

159.  If  two  or  more  denominations  are  of  the  same  kind, 
the  product  of  the  first  by  any  number,  plus  the  product 
of  the  second  by  any  number,  and  so  on,  is  called  a  Com- 
pound Number. 

For  example,  5  miles  +  31  rods  +  4  yards  +  2  feet,  or 
as  it  is  usually  expressed,  5  mi.  31  rd.  4  yd.  2  ft.,  is  a  com- 
pound number. 

Note.  The  number  by  which  the  denomination  is  multiplied  may 
be  an  integer,  a  mixed  number,  or  a  fraction. 

Thus,  3  ft.  5^  in.  is  a  compound  number. 

In  contradistinction,  the  product  of  a  single  denomination 
by  any  number  is  called  a  Simple  Number. 
For  example,  5  mi.  is  a  simple  number. 

160.  Simple  and  Compound  Numbers  are  called  Denomi- 
nate Numbers. 

REDUCTION  OF  DENOMINATE  NUMBERS. 

161.  Reduction  Descending. 

Reduction  Descending  is  the  process  of  expressing  denomi- 
nate numbers  in  terms  of  lower  denominations. 

1.   Express  dB  4  in  pence. 
£4 

20  Since  £1  =  20s.,  £4  =  4  x  20s.,  or  80s. 

80s.  Since  Is.  =  12d.,  80s.  =:  80  x  12d.,  or  960<?. 

12  Therefore,  £  4  =  960(i. 

960d.,  Ans. 


104       .  ARITHMETIC. 

2.   Express  33  rd.  4  yd.  2  ft.  in  inches. 

33  rd. 
5i 


165 

^^i  Since  1  rd.  =  5 J  yd.,  33  rd.  =  33  x  5^  yd.,  or 

181i  yd.  181^  yd.;  then,  181^  yd.  +  4  yd.  =  185^  yd. 

4_  Since  1  yd.  =  3  ft.,  185 J  yd.  =  185^  x  3  ft.,  or 

185i  yd.  556^  ft.;  then,  656^  ft.  +  2  ft.  =  558^  ft. 

3_  Since  1  ft.  =  12  in.,  558^  ft.  =  558|  x  12  in.,  or 

556i-  ft.  6702  in. 

2  Then,  33  rd.  4  yd.  2  ft.  =  6702  in. 


558i  ft. 
12 


6702  in.,  Ans. 

Note.  To  multiply  33  by  6J,  in  Ex.  2,  we  multiply  it  by  5,  and 
then  by  I,  and  add  the  second  result  to  the  first. 

To  multiply  185|  by  3,  we  multiply  185  by  3,  giving  555  ;  we  then 
multiply  ^  by  3,  giving  f ,  or  1^ ;  adding  this  to  555,  the  sum  is  556^. 

3.  Express  f  sq.  yd.  in  square  inches, 
f  sq.  yd.  =  I  X  9  sq.  ft. 

5  ^^ 

=  -  X  9  X  l^  sq.  in.  =  1080  sq.  in.,  Ans. 

P 

4.  Express  1.508  h.  in  seconds. 

1.508  h. 

60 

90.48  min. 
60 


5428.8  sec,  Ans. 

EXAMPLES. 
6.   Express  105  bu.  in  pints. 

6.  How  many  pennyweights  are  there  in  27  lb.  ? 

7.  How  many  inches  are  there  in  35  rd.  ? 

8.  Reduce  5  cd.  to  cubic  inches. 


DENOMINATE   NUMBERS.  105 

9.  Eeduce  127  gal.  3  qt.  to  gills. 

10.  How  many  farthings  are  there  in  £7  15s.  9d  ? 

11.  Express  1  T.  15  cwt.  81  lb.  14  oz.  in  drams. 

12.  Express  78  lb.  9  oz.  13  pwt.  in  grains. 

13.  Express  111b  7  5  53  23  13  gr.  in  grains. 

14.  Eieduce  3  A.  59  sq.  rd.  10  sq.  yd.  to  square  feet. 

15.  Express  365  d.  5  h.  48  min.  49.7  sec.  in  seconds. 

16.  Reduce  2  mi.  253  rd.  3  yd.  2  ft.  11  in.  to  inches. 

17.  Express  13ii  gal.  in  gills. 

18.  Express  ^  T.  in  drams. 

19.  Reduce  fj  °  to  seconds. 

20.  Reduce  ^  mi.  to  inches. 

21.  How  many  seconds  are  there  in  ^  wk.  ? 

22.  How  many  square  inches  are  there  in  ^  A.  ? 

23.  Reduce  £{^  to  farthings. 

24.  Reduce  -^-^  0.  to  minims. 

25.  Express  IO2V2  lb.  troy  in  grains. 

26.  Express  9.53  bu.  in  pints. 

27.  Reduce  .56  wk.  to  seconds. 

28.  How  many  drams  are  there  in  .004992  cwt.  ? 

29.  Reduce  .03581  rd.  to  inches. 

30.  Express  8.604  sq.  rd.  in  square  feet. 

162.  Eeduction  Ascending. 

Reduction  Ascending  is  the  process  of  expressing  denomi- 
nate numbers  in  terms  of  higher  denominations. 

1.   Express  127  gi.  in  terms  of  higher  denominations. 

4)127  gi. Since  4  gi.  =  1  pt.,   there  are  as 

2)31  pt.    +3  gi.  many  pints  in  127  gills  as  4  is  con- 

4)15  qt.   4-lpt.  *^i^^^  ti°^^s  in  127. 

— ^ ,        o — r  4  is  contained  in  127  31  times,  with 

°     *  "^      "  '  the  remainder  3  ;   whence,  127  gi.  = 

3  gal.  3  qt.  1  pt.  3  gi.,  Ans.  31  pt.  3  gi. 


106  ARITHMETIC. 

Since  2  pt.  =  1  qt.,  there  are  as  many  quarts  in  31  pints  as  2  is  con- 
tained times  in  31 ;  that  is,  31  pt.  =15  qt.  1  pt. 

Since  4  qt.  =  1  gal.,  there  are  as  many  gallons  in  15  quarts  as  4  is 
contained  times  in  15  ;  that  is,  15  qt.  =3  gal.  3  qt. 

Whence,  127  gi.  =  3  gal.  3  qt.  1  pt.  3  gi. 

2.   Express  5870  ft.  in  terms  of  higher  denominations. 
3)5870  ft. 

1956  yd. +2  ft. 
2 


11)3912  half-yards. 


3j^0)355  rd.  +  7  half-yards,  or  31  yd.,  or  3  yd.  1 1-  ft. 
1  mi. +  35  rd. 
1  mi.  35  rd.  3  yd.  1^  ft.  -f  2  ft.  =  1  mi.  35  rd.  4  yd.  |  ft.,  Ans. 

Dividing  5870  ft.  by  3,  the  result  is  1956  yd.  2  ft. 

To  reduce  1956  yd.  to  rods,  we  divide  it  by  5i,  or  i^ ;  that  is,  we 
multiply  it  by  2,  and  divide  the  result  by  11  (Art.  104). 

Now  1956  yd.  x  2  =  3912  half-yards;  dividing  this  by  11,  the  result 
is  355  rd.  and  7  half-yards. 

Also,  7  half -yards  =  3i  yd.  =  3  yd.  li  ft. 

Dividing  355  rd.  by  320,  we  have  1  mi.  35  rd. 

Therefore,  5870  ft.  =  1  mi.  35  rd.  3  yd.  li  ft.  -)-  2  f t. 

But  1  ft.  +  2  ft.  =  3  ft.  =  1  yd. 

Whence,  5870  ft.  =  1  mi.  35  rd.  4  yd.  i  ft. 

EXAMPLES. 
Express  in  terms  of  higher  denominations  : 


3. 

975  pt.  (dry  meas.) 

13. 

49326  gr.  (troy.) 

4. 

2367  oz.  (av.) 

14. 

45342  in. 

5. 

6571  far. 

15. 

113292  sq.  ft. 

6. 

8027  gr.  (apoth.) 

16. 

800000  sec. 

7. 

35798  m.. 

17. 

525086  cu.  in. 

8. 

1523  ft. 

18. 

951000  dr. 

9. 

2609  gi. 

19. 

334476  in. 

10. 

29760  far. 

20. 

41253  sq.  in. 

11. 

323515  sec. 

21. 

70000  in. 

12. 

123549". 

22. 

89582  sq.  ft. 

DENOMINATE   NUMBERS.  107 


ADDITION  OF  DENOMINATE   NUMBERS. 

163.   1.  Find  the  sum  of   17  lb.  10  oz.  15  pwt.  19  gr., 
12  lb.  9  oz.  19  pwt.  20  gr.,  and  8  lb.  5  oz.  11  pwt.  15  gr. 

17  1b.     10  oz.     15  pwt.     19  gr. 
12  9  19  20 

8  5  11  15 


39  lb.       2  oz.       7  pwt.       6  gr.,  Ans. 

The  sum  of  15  gr.,  20  gr.,  and  19  gr.,  is  54  gr.,  or  2  pwt.  6  gr. ;  we 
then  write  6  gr.  under  the  column  of  grains. 

The  sum  of  2  pwt.,  11  pwt,  19  pwt.,  and  15  pwt.,  is  47  pwt.,  or 
2  oz.  7  pwt.  ;  we  then  write  7  pwt.  under  the  column  of  pennyweights. 

The  sum  of  2  oz.,  5  oz.,  9  oz.,  and  10  oz.,  is  26  oz.,  or  2  lb.  2  oz.  ; 
we  then  write  2  oz.  under  the  column  of  ounces. 

The  sum  of  2  lb.,  8  lb.,  12  lb.,  and  17  lb.,  is  39  lb. ;  we  then  write 
39  lb.  under  the  column  of  pounds. 

The  final  result  is  39  lb.  2  oz.  7  pwt.  6  gr. 

2.  Add  3  mi.  201  rd.  3  yd.  2  ft.  9  in.,  4  mi.  76  rd.  5  yd. 
0  ft.  11  in.,  2  mi.  253  rd.  4  yd.  2  ft.  5  in.,  and  8  mi.  189  rd. 
5  yd.  1  ft.  8  in. 


3  mi. 

201  rd. 

3  yd. 

2  ft. 

9  in. 

4 

76 

5 

0 

11 

2 

253 

4 

2 

5 

8 

189 

5 

1 

8 

19  mi. 

82  rd. 

2iyd. 

1ft. 
1 

9  in. 
6 

19  mi.       82  rd.     3    yd.     0  ft.     3  in.,  Ans. 

The  sum  of  the  inches  is  33  in.,  or  2  ft.  9  in. 
The  sum  of  the  feet  is  7  ft. ,  or  2  yd.  1  ft. 
The  sum  of  the  yards  is  19  yd.,  or  3  rd.  2^  yd. 
The  sum  of  the  rods  is  722  rd.,  or  2  mi.  82  rd. 
The  sum  of  the  miles  is  19  mi. 

But  J  yd.  =  1  ft.  6  in. ;  which,  added  to  19  mi.  82  rd.  2  yd.  1  ft. 
9  in.,  gives  19  mi.  82  rd.  3  yd.  0  ft.  3  in. 


108  ARITHMETIC. 

EXAMPLES. 
Add  the  following : 

3.  12  gal.  3  qt.  1  pt.  2  gi.,  7  gal.  1  qt.  0  pt.  1  gi.,  and  24 
gal.  2  qt.  1  pt.  2  gi. 

4.  4  cwt.  67  lb.  8  oz.  14  dr.,  11  cwt.  49  lb.  15  oz.  9  dr., 
and  2  cwt.  83  lb.  12  oz.  13  dr. 

5.  34  bu.  3  pk.  7  qt.  1  pt.,  46  bu.  1  pk.  5  qt.  1  pt.,  55  bu. 
2  pk.  6  qt.  0  pt.,  and  27  bu.  3  pk.  4  qt.  1  pt. 

6.  71  d.  22  h.  18  min.  45  sec,  36  d.  16  h.  48  min.  30  sec., 
60  d.  11  h.  32  min.  12  sec,  49  d.  9  h.  23  min.  54  sec,  and 

56  d.  17  h.  57  min.  29  sec. 

7.  45°  38' 40",  123°  17' 33",  78°  44' 55",  and  65°  46' 18". 

8.  211b85  53l3  llgr.,  321b  105  7  3  2  3  18  gr.,  81b 
9563237  gr.,  and  16  lb  65  4  3  1  3  14  gr. 

9.  7  mi.  163  rd.  2  yd.  2  ft.  11  in.,  2  mi.  313  rd.  5  yd.  1  ft. 
7  in.,  9  mi.  36  rd.  4  yd.  2  ft.  8  in.,  and  6  mi.  244  rd.  3  yd. 
1  ft.  10  in. 

10.  £  25  16s.  Sd.  2  far.,  £  86  6s.  lid.  3  far.,  £  74  18s.  9d, 
£100  15s.  lOd  1  far.,  £64  9s.  6d.  3  far.,  and  £98  12s.  7d. 

11.  13  T.  19  cwt.  81  lb.  9  oz.  14  dr.,  8  T.  11  cwt.  64  lb. 
12  oz.  7  dr.,  21  T.  16  cwt.  92  lb.  11  oz.  10  dr.,  15  T.  9  cwt. 

57  lb.  8  oz.  13  dr.,  and  9  T.  15  cwt.  70  lb.  15  oz.  9  dr. 

12.  45  cd.  123  cu.  ft.  828  cu.  in.,  26  cd.  64  cu.  ft.  1457  cu. 
in.,  and  73  cd.  98  cu.  ft.  585  cu.  in. 

13.  6  lb.  11  oz.  14  pwt.  15  gr.,  13  lb.  5  oz.  16  pwt.  8  gr., 
28  lb.  8  oz.  13  pwt.  22  gr.,  and  19  lb.  6  oz.  15  pwt.  13  gr. 

14.  3  sq.  mi.  288  A.  59  sq.  rd.  19  sq.  yd.  8  sq.  ft.  74  sq. 
in.,  5  sq.  mi.  446  A.  115  sq.  rd.  23  sq.  yd.  6  sq.  ft.  109  sq.  in., 
and  4  sq.  mi.  610  A.  149  sq.  rd.  28  sq.  yd.  7  sq.  ft.  141  sq.  in. 

16.  12  mi.  98  rd.  5  yd.  1  ft.  8  in.,  9  mi.  212  rd.  4  yd.  2  ft. 
10  in.,  21  mi.  156  rd.  5  yd.  2  ft.  8  in.,  18  mi.  303  rd.  3  yd. 
1  ft.  9  in.,  36  mi.  286  rd.  4  yd.  2  ft.  11  in.,  and  30  mi.  123 
rd.  5  yd.  1  ft.  10  in. 


DENOMINATE  NUMBERS.  109 


SUBTRACTION  OF  DENOMINATE   NUMBERS. 

164.  1.  Subtract  5  T.  18  cwt.  43  lb.  from  14  T.  12  cwt. 
69  1b. 

14  T.  12  cwt.  69  lb.  43  lb.  from  69  lb.  leaves  26  lb. 

5         18  43  Since   18  cwt.   is  greater  than  12 

cwt.,  we  take  1  T.,  or  20  cwt.,  from 

8  T.  14  cwt.  26  lb.,  Ans.     the  14  T.  of  the  minuend,  leaving  13 

T. ;   adding  20  cwt.  to  12  cwt.,  we 
have  32  cwt. ;  then,  18  cwt.  from  32  cwt.  leaves  14  cwt. 
Finally,  5  T.  from  13  T.  leaves  8  T. 

2.  Subtract  2  A.  103  sq.  rd.  10  sq.  yd.  8  sq.  ft.  23  sq.  in. 
from  3  A.  100  sq.  rd.  10  sq.  yd. 

3  A.  100  sq.  rd.  10  sq.  yd.  0  sq.  ft.   0  sq.  in. 
2   103      10       8      23 

156  sq.  rd.  29J  sq.  yd.  0  sq.  ft.  121  sq.  in. 
2      36 

156  sq.  rd.  29  sq.  yd.  3  sq.  ft.  13  sq.  in.,  Ans. 

We  take  1  sq.  yd.,  or  9  sq.  ft.,  or  8  sq.  ft.  144  sq.  in.  from  the  10 
sq.  yd.  of  the  minuend ;  then,  23  sq.  in.  from  144  sq.  in.  leaves  121  sq. 
in.,  and  8  sq.  ft.  from  8  sq.  ft.  leaves  0  sq.  ft. 

10  sq.  yd.  from  39^  sq.  yd.  leaves  29^  sq.  yd.,  and  103  sq.  rd.  from 
259  sq.  rd.  leaves  156  sq.  rd. 

But  I  sq.  yd.  =  2  sq.  ft.  36  sq.  in.  ;  which,  added  to  156  sq.  rd.  29 
sq.  yd.  0  sq.  ft.  121  sq.  in.,  gives  156  sq.  rd.  29  sq.  yd.  3  sq.  ft. 
13  sq.  in. 

EXAMPLES. 

Subtract  the  following : 

3.  2  bu.  3  pk.  7  qt.  from  13  bu.  2  pk.  5  qt.  1  pt. 

4.  £87  13s.  9d.  1  far.  from  £105  13s.  Sd.  3  far. 

5.  49°  52'  41"  from  87°  39'  25". 

6.  38  gal.  3  qt.  1  pt.  3  gi.  from  48  gal.  3  qt.  1  pt.  2  gi. 

7.  42  lb.  7  oz.  18  pwt.  4  gr.  from  52  lb.  7  oz. 

8.  5  cu.  yd.  26  cu.  ft.  752  cu.  in.  from  13  cu.  yd.  9  cu.  ft. 


110  ARITHMETIC. 

9.  14  0.  13  f  5  2f3  48rTLfrom24  0.  4f5  7f3  23ni. 

10.  7  mi.  116  rd.  4  yd.  2  ft.  from  15  mi.  0  rd.  3  yd.  1  ft. 

11.  14  T.  16  cwt.  81  lb.  13  oz.  15  dr.  from  15  T.  11  cwt. 

12.  9  A.  147  sq.  rd.  19  sq.  yd.  5  sq.  ft.  from  15  A. 

13.  1  d.  20  h.  31  min.  56  sec.  from  6  d.  15  h.  48  min. 

14.  ll'  mi.  250  rd.  2  yd.  2  ft.  3  in.  from  28  mi.  45  rd. 
2  yd.  1  ft.  9  in. 

16.  4  A.  152  sq.  rd.  28  sq.  yd.  7  sq.  ft.  130  sq.  in.  from 
23  A.  153  sq.  rd.  5  sq.  yd.  8  sq.  ft.  103  sq.  in. 

165.  To  Find  the  Difference  in  Time  between  Two  Dates.' 

To  find  the  exact  number  of  days  between  two  dates,  we 
proceed  as  follows : 

1.  Find  the  exact  number  of  days  from  July  12  to  Oct.  5. 
19  The  number  of  days  left  in  July  is  19. 

31  The  number  of  days  in  August  is  31,  and  in  Septem- 

30  ber,  30. 

5  From  Oct.  1  to  Oct.  5  is  5  days.      , 

85  d.   Ans.       Adding,  the  required  number  of  days  is  85. 

Allowance  must  be  made  for  leap  years  in  finding  the  exact 
number  of  days  between  two  dates. 

2.  Find  the  exact  number  of  days  from  Feb.  17,  1880,  to 
May  9,  1888. 

Since  1880  and  1884  are  leap  years,  the  number  of  days  from  Feb. 
17,  1880,  to  Feb.  17,  1888,  is  8  x  365,  plus  2,  or  2922. 

Since  1888  is  a  leap  year,  the  number  of  days  from  Feb.  17,  1888, 
to  May  9,  1888,  is  12  +  31  +  30  +  9,  or  82. 

Then,  2922  d.  +  82d.  =  3004  d.,  Ans. 

To  find  the  time  between  two  dates  in  years,  months,  and 
days,  the  following  method  is  employed  by  business  men : 

3.  Find  the  time  from  June  15,  1887,  to  April  8,  1892. 

From  June  15,  1887,  to  June  15,  1891,  is  4  y. 
From  June  15,  1891,  to  March  15,  1892,  is  9  mo. 
From  March  16,  1892,  to  April  8,  1892,  the  exact  number  of  days  is 
16  +  8,  or  24. 

Then  the  required  result  is  4  y .  9  mo.  24  d. ,  Ans. 


DENOMINATE   NUMBERS.  HI 

Note.  In  reckoning  by  the  above  method  from  the  31st  day  of  a 
month  to  the  31st  day  of  a  month  not  having  31  days,  we  reckon  to 
the  last  day  of  the  latter  month. 

Thus,  to  find  the  time  from  March  31  to  Oct.  17,  we  call  from  March 
31  to  Sept.  30  6  mo.,  and  from  Sept.  30  to  Oct.  17  17  d.  ;  the  result  is 
6  mo.  17  d. 

Again,  to  find  the  time  from  Jan.  28  to  March  4,  in  an  ordinary 
year,  we  call  from  Jan.  28  to  Feb.  28  1  mo.,  and  from  Feb.  28  to 
March  4  4  d.  ;  the  result  is  1  mo.  4  d. 

But  in  a  leap  year,  from  Jan.  28  to  Feb.  28  is  1  mo.,  and  from  Feb. 
28  to  March  4  is  5  d.  ;  in  this  case,  the  time  is  1  mo.  5  d. 

EXAMPLES. 

Find  the  exact  number  of  days  from : 

4.  March  27,  1886,  to  Dec.  10,  1886. 

5.  April  11,  1891,  to  Jan.  31,  1892. 

6.  Sept.  14,  1887,  to  June  5,  1888. 

7.  July  4,  1889,  to  May  30,  1891. 

8.  Nov.  20,  1887,  to  Aug.  8,  1889. 

9.  Feb.  18,  1883,  to  Oct.  1, 1892. 
Find  the  time  in  years,  months,  and  days  from : 

10.  March  13,  1885,  to  Jan.  29,  1893. 

11.  Feb.  19,  1884,  to  June  8,  1889. 

12.  April  30,  1888,  to  March  22,  1890. 

13.  Aug.  31,  1881,  to  May  5,  1887. 

14.  Dec.  16,  1883,  to  Dec.  1,  1892. 

15.  Oct.  28,  1886,  to  March  18,  1888. 

MULTIPLICATION  OF  DENOMINATE  NUMBERS. 

•    166.   1.  Multiply  2  mi.  135  rd.  5  yd.  2  ft.  10  in.  by  7. 
2  mi.     135  rd.     5    yd.     2  ft.     10  in. 


16  mi.     312  rd.     2^  yd.     1ft.     10  in. 

1  6 

16  mi.     312  rd.     3    yd.     0  ft.       4  in.,  Ans. 


112  ARITHMETIC. 

7  X  10  in.  is  70  in.,  or  5  ft.  10  in. 
7  X  2  ft.,  plus  5  ft.,  is  19  ft.,  or  6  yd.  1  ft.. 
7  X  5  yd.,  plus  6  yd.,  is  41  yd.,  or  7  rd.  2 J  yd. 
7  X  135  rd.,  plus  7  rd.,  is  962  rd.,  or  2  mi.  312  rd. 
7  X  2  mi.,  plus  2  mi.,  is  16  mi. 

But  ^  yd.  =  1  ft.  6  in. ;  which,  added  to-  16  mi.  312  rd.  2  yd.  1  ft. 
10  in.,  gives  16  mi.  312  rd.  3  yd.  0  ft.  4  in. 

EXAMPLES. 
Multiply  the  following : 

2.  13  bu.  3  pk.  5  qt.  1  pt.  by  3. 

3.  26°  38' 51.32"  by  8. 

4.  25  gal.  2  qt.  1  pt.  3|  gi.  by  6. 

5.  8  0.  13fS  5f3  31ni  by  7. 

6.  18  mi.  275  rd.  5  yd.  2  ft.  by  4. 

7.  5  d.  19  h.  37  min.  42  sec.  by  15. 

8.  £  86  16s.  9d.  1|  far.  by  18. 

9.  16  T.  9  cwt.  82  lb.  10  oz.  14.5  dr.  by  5. 

10.  3  cd.  87  cu.  ft.  561  cu.  in.  by  11. 

11.  12  lb.  7  oz.  18  pwt.  13  gr.  by  20. 

12.  213  rd.  1  yd.  2  ft.  6  in.  by  9. 

13.  5  A.  97  sq.  rd.  22  sq.  yd.  4  sq.  ft.  75  sq.  in.  by  10. 

DIVISION  OP  DENOMINATE  NUMBERS. 

167.  1.  Divide  46  A.* 52  sq.  rd.  21  sq.  yd.  6  sq.  ft.  30  sq. 

in.  by  6. 

6)46  A.  52  sq.  rd.  21  sq.  yd.  6  sq.  ft.  30  sq.  in. 

7  A.  115  sq.  rd.  13  sq.  yd.  6  sq.  ft.  41  sq.  in.,  Ans. 

46  A.  -r-  6  =  7  A. ,  with  a  remainder  of  4  A. ,  or  640  sq.  rd. 

640  sq.  rd.  +  52  sq.  rd.  =  692  sq.  rd.  ;  692  sq.  rd.  -i-  6  =  115  sq.  rd., 
with  a  remainder  of  2  sq.  rd. ,  or  60^  sq.  yd. 

60^  sq.  yd.  +  21  sq.  yd.  =  81 1  sq.  yd.  ;  811  sq.  yd.  -r-  6  =  13  sq.  yd., 
with  a  remainder  of  3^  sq.  yd.,  or  31^  sq.  ft. 


DENOMINATE  NUMBERS.  113 

31^  sq.  ft.  +  6  sq.  ft.  =  37^  sq.  ft.  ;  37i  sq.  ft.-^  6  =  6  sq.  ft.,  with  a 
remainder  of  li  sq.  ft.,  or  216  sq.  in. 

216  sq.  in.  +  30  sq.  in.  =  246  sq.  in.  ;  246  sq.  in.  -r-  6  =  41  sq.  in. 
Then  the  result  is  7  A.  115  sq.  rd.  13  sq.  yd.  6  sq.  ft.  41  sq.  in. 

EXAMPLES. 
Divide  the  following : 

2.  35  bu.  3  pk.  6  qt.  1  pt.  by  3. 

3.  60  gal.  0  qt.  1  pt.  1  gi.  by  11. 

4.  22°  30'  by  8. 

5.  234  T.  17  cwt.  89  lb.  6  oz.  by  10. 

6.  128  cu.  yd.  1  cu.  ft.  759  cu.  in.  by  5. 

7.  40  lb.  8  oz.  8  pwt.  15J  gr.  by  7. 

8.  273  rd.  4  yd.  0  ft.  10  in.  by  2. 

9.  £  147  95.  9d.  2  far.  by  15. 

10.  451  gal.  1  qt.  0  pt.  3  gi.  by  13. 

11.  3  wk.  5  d.  6  h.  46  min.  12.8  sec.  by  4. 

12.  57  lb  105  2  3  13  4  gr.  by  12. 

13.  19  mi.  91  rd.  2  yd.  1  ft.  6  in.  by  9. 

14.  22  A.  154  sq.  rd.  4  sq.  yd.  8  sq.  ft.  114  sq.  in.  by  6. 

168.  To  Multiply  or  Divide  a  Denominate  Number  by  a 
Fraction. 

1.   Multiply  8  T.  10  cwt.  73  lb.  5  oz.  11  dr.  by  |. 

We  multiply  the  given  denominate  number  by  3,  and 
divide  the  result  by  4. 

8  T.  10  cwt.  73  lb.  5  oz.  11    dr. 
3 


4)25  T.  12  cwt.  20  lb.  1  oz.    1    dr. 

6  T.    8  cwt.    5  lb.  0  oz.    4-J  dr.,  Ans. 

To  divide  a  denominate  number  by  a  fraction,  multiply  it 
by  the  denominator,  and  divide  the  result  by  the  numerator. 


114  ARITHMETIC. 

To  multiply  a  denominate  number  by  a  mixed  number, 
multiply  it  by  the  integer  and  the  fraction  separately,  and 
add  the  results. 

To  divide  a  denominate  number  by  a  mixed  number,  the 
divisor  should  be  expressed  in  a  fractional  form. 

To  multiply  or  divide  a  denominate  number  by  a  decimal, 
the  latter  should  be  expressed  as  a  common  fraction. 

EXAMPLES. 

2.  Multiply  £  14  16s.  9d.  3  far.  by  y. 

3.  Multiply  13  lb.  10  oz.  15  pwt.  9  gr.  by  3^. 

4.  Divide  2  T.  3  cwt.  46  lb.  13  oz.  6  dr.  by  f . 

5.  Divide  14  bu.  3  pk.  2  qt.  1  pt.  by  2f . 

6.  Divide  2  d.  9  h.  8  min.  42  sec.  by  f . 

7.  Multiply  7  mi.  210  rd.  4  yd.  2  ft.  9  in.  by  .4. 

8.  Divide  2  gal.  2  qt.  1  pt.  2.45  gi.  by  .35. 

9.  Multiply  12  cd.  25  cu.  ft.  1365  cu.  in.  by  f 

10.  Divide  38°  58' 16"  by  1.7. 

11.  Multiply  6  ft)  55  43  03  10  gr.  by  2.5. 

169.  To  Divide  One  Denominate  Number  by  Another  of 
the  Same  Kind. 

To  divide  one  denominate  number  by  another  of  the  same 
kind,  express  them  in  terms  of  the  same  denomination,  and 
divide  the  results. 

1.   Divide  141  bu.  1  pk.  7  qt.  by  7  bu.  3  pk.  3  qt.  1  pt. 

The  lowest  denomination  expressed  in  either  of  the  given 

numbers  is  pints  ;   reducing  both  dividend  and  divisor  to 

pints,  we  have 

141  bu.  1  pk.  7  qt.  =  9054  pt., 

and  7  bu.  3  pk.  3  qt.  1  pt.  =    503  pt. 

503)9054(18,  Ans. 
503 
4024 
4024 


DENOMINATE   NUMBERS.  115 

EXAMPLES. 
Divide  : 

2.  £  35  5s.  Sd.  3  far.  by  £  1  8s.  2d.  3  far. 

3.  266  gal.  1  qt.  1  pt.  1  gi.  by  8  gal.  2  qt.  0  pt.  3  gi. 

4.  52  T.  13  cwt.  87  lb.  by  6  T.  11  cwt.  73  lb.  6  oz. 

5.  69  d.  21  h.  54  min.  48  sec.  by  5  d.  19  h.  49  min.  34  sec. 

6.  10  mi.  183  rd.  4  yd.  0  ft.  3  in.  by  260  rd.  1  yd.  1  ft.  9  in. 

TO  EXPRESS  A  FRACTION  OR   DECIMAL   OF  A  SIMPLE 
NUMBER  IN  LOWER  DENOMINATIONS. 

170.   1.   Express  j^  mi.  in  lower  denominations. 

f  mi.  =  f  X  320  rd.  =  H^  rd.  =  228f  rd. 

f  rd.  =f  X  Yyd.   =¥yd.     =3}  yd. 

1  yd.  =}  X  3  ft.       =  f  ft. 

f  ft.    =  f  X  12  in.    =  -3j6  in.      =  5f  in. 
Hence,  f  mi.  =  228  rd.  3  yd.  0  ft.  5^  in.,  Ans. 

2.  Express  .89  gal.  in  lower  denominations. 

.89  gal. 

4 

3.56  qt.  -89  gal.  =  .89  x  4  qt.,  or  3.56  qt. 

2  .56  qt.    =  .56  X  2  pt.,  or  1.12  pt. 

J32pt.  -^^P*-    =.12x4gi.,  or0.48gi. 

^  Then,  the  required  result  is  3  qt.  1  pt. 

aTo     •  0.48  gi. 

0.48  gi.  ^ 

3  qt.  1  pt.  0.48  gi.,  Ans. 


EXAMPLES. 

Express  in  lower  denominations : 

3.  |wk. 

8.    .605  T. 

13. 

^A- 

4.  3^  lb.  troy. 

9.  AO. 

14. 

Acd. 

6.    Hgal. 

10.   frd. 

15. 

.51  mi. 

6.   .3bu. 

11.   .7  sq.  rd. 

16. 

£  .0955. 

7.   .871°. 

12.   .293  R). 

17. 

^A. 

116  ARITHMETIC. 

TO  EXPRESS  A  DENOMINATE  NUMBER  AS  A  FRACTION 
OR  DECIMAL  OF  A  SINGLE  DENOMINATION. 

171.   1.   Express  71  rd.  0  yd.  1  ft.  10  in.  as  a  fraction  of 

a  mile. 

10  in.    =fft. 

If  ft.     =(-V--^3)yd.,ori|yd. 

iiyd.  =(H--¥-)rd.,orird. 

71i  rd.  =  i^^-i-  320)  mi.,  or  f  mi.,  Ans. 

2.   Express  2  d.  4  h.  4  min.  48  sec.  as  a  decimal  of  a 

week. 

48  sec.     =  f  min.  =  .8  min. 


4.8  min.  =  (4.8 
4.08  h.  =  (4.08 
2.17  d.    =(2.17 


60)  h.  =  .08  h. 
24)  d.  =  .17  d. 
7)  wk.  =  .31  wk.,  Ans. 


EXAMPLES. 
Express : 

3.  10  oz.  13  pwt.  8  gr.  as  a  fraction  of  a  pound. 

4.  3s.  Id.  2  far.  as  a  fraction  of  a  pound. 

5.  30'  24"  as  a  fraction  of  a  degree. 

6.  2  pk.  2  qt.  1  pt.  as  a  fraction  of  a  bushel. 

7.  16  h.  19  min.  12  sec.  as  a  fraction  of  a  day. 

8.  2  qt.  1  pt.  1^  gi.  as  a  fraction  of  a  gallon. 

9.  69  cu.  ft.  576  cu.  in.  as  a  fraction  of  a  cord. 

10.  4  yd.  1  ft.  2^  in.  as  a  fraction  of  a  rod. 

11.  71  lb.  6  oz.  13|^  dr.  as  a  fraction  of  a  hundred-weight. 

12.  5  f  5  0  f  3  57f  K  as  a  fraction  of  a  pint. 

13.  43  sq.  rd.  19  sq.  yd.  2  sq.  ft.  36  sq.  in.  as  a  fraction 
of  an  acre. 

14.  3  pk.  1  qt.  1^  pt.  as  a  decimal  of  a  bushel. 

15.  25'  39"  as  a  decimal  of  a  degree. 

16.  3  qt.  1  pt.  3  gi.  as  a  decimal  of  a  gallon. 

17.  4  d.  19  h.  55  min.  12  sec.  as  a  decimal  of  a  week. 


DENOMINATE  NUMBERS.         *  117 

18.  2s.  lid.  2.08  far.  as  the  decimal  of  a  pound. 

19.  16  cu.  ft.  1512  cu.  in.  as  a  decimal,  of  a  cubic  yard. 

20.  10  oz.  5  pwt.  4.8  gr.  as  a  decimal  of  a  pound. 

21.  15  cwt.  6  lb.  6  oz.  6.4  dr.  as  a  decimal  of  a  ton. 

22.  9  sq.  yd.  0  sq.  ft.  97.2  sq.  in.  as  a  decimal  of  a  square 
rod. 

23.  259  rd.  1  yd.  0  ft.  3.6  in.  as  a  decimal  of  a  mile. 

TO  EXPRESS  ONE  DENOMINATE  NUMBER  AS  A  FRAC- 
TION OR  DECIMAL  OF  ANOTHER. 

172.  To  express  one  denominate  number  as  a  fraction  or 
decimal  -of  another,  reduce  them  to  the  same  denomination, 
and  divide  the  results. 

1.  Express  5  gal.  1  qt.  1  pt.  3  gi.  as  a  fraction  of  7  gal. 

2  qt.  1  pt.  1  gi. 

5  gal.  1  qt.  1  pt.  3  gi.  =  175  gi. 
7  gal.  2  qt.  1  pt.  1  gi.  =  245  gi. 

iH  =  f,Ans. 

2.  Express  2  sq.  rd.  21  sq.  yd.  8^  sq.  ft.  as  a  decimal  of 

3  sq.  rd.  6  sq.  yd.  4J  sq.  ft. 

2  sq.  rd.  21  sq.  yd.  81  sq.  ft.  =  742  sq.  ft. 

3  sq.  rd.    6  sq.  yd.  4J  sq.  ft.  =  875  sq.  ft. 

iH  =  {i^  =  MS,Ans. 

EXAMPLES. 

Express  : 

3.  1  bu.  3  pk.  4  qt.  1  pt.  as  a  fraction  of  3  bu.  3  pk.  1  qt, 

4.  1  d.  13  h.  30  min.  as  a  fraction  of  5  d. 

5.  £  2  5s.  lie?.  1  far.  as  a  fraction  of  £  4  18s.  5d.  1  far. 

6.  2°  46'  30''  as  a  fraction  of  3°  4'  30". 

7.  4  T.  13  cwt.  33^  lb.  as  a  fraction  of  7  T. 


118  ARITHMETIC. 

8.  6  rd.  1  yd.  1  ft.  4  in.  as  a  fraction  of  7  rd.  4  yd.  If  ft. 

9.  1  S  4  3  0  3  15  gr.  as  a  fraction  of  2  5  6  3  0  3  3  gr. 

10.  1  lb.  5  oz.  1  pwt.  8  gr.  as  a  fraction  of  1  lb.  11  oz. 
9  pwt.  8  gr. 

11.  45  sq.  rd.  27  sq.  yd.  5f  sq.  ft.  as  a  fraction  of  80  sq. 
rd.  10  sq.  yd.  5  sq.  ft. 

12.  2  gal.  3  qt.  0  pt.  3  gi.  as  a  decimal  of  5  gal.  1  qt.  1  pt. 
3gi. 

13.  5  bu.  2  pk.  2  qt.  1  pt.  as  a  decimal  of  7  bn.  3  pk.  7  qt. 

14.  2  lb.  5  oz.  9  dr.  as  a  decimal  of  2  lb.  7  oz.  1  dr. 

15.  5  cu.  ft.  1440  cu.  in.  as  a  decimal  of  7  cu.  ft.  1344 
cu.  in. 

16.  1  h.  54  min.  20  sec.  as  a  decimal  of  2  h.  10  min. 
40  sec. 

17.  3  lb.  6  oz.  7  pwt.  22  gr.  as  a  decimal  of  5  lb.  7  oz. 
16  pwt.  16  gr. 

18.  2  mi.  310  rd.  2.2  yd.  as  a  decimal  of  11  mi. 

19.  61  sq.  rd.  21  sq.  yd.  3f  sq.  ft.  as  a  decimal  of  154  sq. 
rd.  8  sq.  yd.  1^-  sq.  ft. 

LONGITUDE  AND  TIME. 

173.  Since  the  earth  revolves  upon  its  axis  once  in  24 
hours,  if  the  sun  crosses  the  meridian  of  any  place  at  a 
certain  time,  it  will  cross  a  meridian  1°  to  the  west  of  the 
first  -^  of  24  hours,  or  ^  of  an  hour,  later;  that  is,  4 
minutes  later. 

Hence,  the  local  time  for  places  on  the  first  meridian  is 
4  minutes  faster  than  for  places  on  the  second. 

In  like  manner,  the  local  time  on  the  first  meridian  is : 
1  h.  faster  than  on  a  meridian  15°  to  the  west ; 
1  min.  faster  than  on  a  meridian  15'  to  the  west ; 
1  sec.  faster  than  on  a  meridian  15"  to  the  west ; 
4  sec.  faster  than  on  a  meridian  1'  to  the  west ; 
•^  sec.  faster  than  on  a  meridian  1"  to  the  west. 


DENOMINATE  NUMBERS.  119 

1.  The  longitude  of  Boston  is  71°  4'  W,,  and  of  Kome, 
12°  27'  E. ;  what  is  the  difference  in  time  between  the  cities  ? 

The  difference  in  longitude  is  71° 4*  +  12° 27',  or  83° 31'. 

The  difference  in  time  for  1°  being  4  min.,  for  83°  it  is  83  x  4  min.; 
that  is,  332  min.,  or  5  h.  32  min. 

The  difference  in  time  for  1'  being  4  sec,  for  31'  it  is  31  x  4  sec; 
that  is,  124  sec,  or  2  min.  4  sec. 

Therefore  the  difference  in  time  is 

5  h.  32  min.  +  2  min.  4  sec,  or  5  h.  34  min.  4  sec,  Ans. 

Note.  If  the  sum  of  the  longitudes  of  two  places,  in  E.  and  W. 
longitude,  respectively,  is  greater  than  180°,  the  difference  in  longitude 
may  be  found  by  subtracting  this  sum  from  360°. 

2.  If  the  local  time  at  Paris  is  5  h.  43  min.  56  sec.  slower 
than  at  Calcutta,  what  is  the  difference  in  longitude  between 
the  two  places  ? 

The  difference  in  longitude  for  1  h.  being  15°,  for  5  h.  it  is  5  x  15°, 
or  75°. 

The  difference  in  longitude  for  1  min.  being  15',  for  43  min.  it  is 
43  X  15';  that  is,  645',  or  10°  45'. 

The  difference  in  longitude  for  1  sec.  being  15",  for  56  sec.  it  is 
56  X  15";  that  is,  840",  or  14'. 

Then  the  difference  in  longitude  is  75°  +  10°  45'  +  14',  or  85°  59',  Ans. 

EXAMPLES. 

Find  the  difference  in  time  between : 

3.  New  York,  Ion.  74°  0'  W.,  and  San  Francisco,  Ion. 
122°  27' W. 

4.  St.  Petersburg,  Ion.  30°  19'  E.,  and  Valparaiso,  Ion. 
71°42' W. 

5.  Calcutta,  Ion.  88°19'2"  E.,  and  St.  Paul,  Ion.  93°4'55"  W. 
S.   Berlin,  Ion.  13°24'28"  E.,  and  Sydney,  Ion.  152°19'37"E. 

7.  When  it  is  10.30  p.m.  at  Washington,  Ion.  77°  3'  W., 
what  time  is  it  at  Constantinople,  Ion.  28°  49'  E.  ? 

8.  When  it  is  3.15  p.m.  at  Paris,  Ion.  2°  20' 17"  E.,  what 
time  is  it  at  New  Orleans,  Ion.  90°  2'  28"  W.  ? 


120  ARITHMETIC. 

9.  When  it  is  11.35  a.m.  at  Pekin,  Ion.  116°  24'  17"  E., 
what  time  is  it  at  Chicago,  Ion.  87°  34'  53"  W.  ? 

10.  When  it  is  6.27  a.m.  at  Canton,  Ion.  113°  15' 33"  E., 
what  time  is  it  at  Greenwich,  England  ? 

Find  the  difference  in  longitude  between  two  places  whose 
difference  in  time  is  : 

11.  5  h.  57  min.  13.   11  h.  13  min.  5  sec. 

12.  2  h.  34  min.  42  sec.  14.   8  h.  45  min.  38  sec. 

15.  The  local  time  at  Sandy  Hook  is  4  h.  56  min.  4  sec. 
slower  than  that  at  Greenwich,  England ;  what  is  the  longi- 
tude of  Sandy  Hook  ? 

16.  When  it  is  4.25  p.m.  at  Kome,  it  is  9  h.  42  min. 
24  sec.  a.m.  at  Mobile ;  if  the  longitude  of  Eome  is  12°  27' 
E.,  what  is  the  longitude  of  Mobile  ? 

17.  When  it  is  1.20  a.m.  at  St.  Louis,  Ion.  90°  15'  15"  W., 
it  is  8  h.  35  min.  If  sec.  a.m.  at  the  Cape  of  Good  Hope. 
What  is  the  longitude  of  the  Cape  of  Good  Hope  ? 

18.  When  it  is  6.33  p.m.  at  Jerusalem,  Ion.  35°  30' 48"  E., 
it  is  11  h.  17  min.  13  sec.  a.m.  at  Montreal.  What  is  the 
longitude  of  Montreal  ? 

19.  When  it  is  3.55  a.m.  at  Constantinople,  Ion.  28°  58' 
40"  E.,  it  is  6  h.  50  min.  38  sec.  a.m.  at  Bombay.  What  is 
the  longitude  of  Bombay  ? 

PROBLEMS. 

174.  1.  How  many  lots,  each  containing  1  A.  36  sq.  rd.,  can 
be  made  from  a  piece  of  land  containing  20  A.  132  sq.  rd.  ? 

2.  What  is  the  value  of  18  T.  11  cwt.  20  lb.  of  coal,  at 
$  5.25  a  ton  ? 

3.  Which  is  the  faster,  a  train  which  runs  225  rods  a 
minute,  or  one  which  runs  a  mile  in  85  seconds  ? 

4.  If  a  man  can  do  a  piece  of  work  in  7  h.  34  min.  10 
sec,  how  long  will  it  take  five  men  to  do  the  work  ? 


DENOMINATE  NUMBERS.  121 

5.  A  note  dated  Oct.  15,  1889,  was  paid  July  3,  1891. 
How  long  did  it  run  ? 

6.  A  gentleman  divided  his  estate  of  35  acres  equally 
between  his  six  children.     How  much  did  each  receive  ? 

7.  The  sides  of  a  field  are,  respectively,  8  rd.  1  yd.  2  ft. 
7  in.,  9  rd.  1  yd.  2  ft.  6  in.,  12  rd.  1  yd.  0  ft.  11  in.,  and 
10  rd.  1  yd.  2  ft.  3  in.  What  is  the  distance  around  the 
field? 

8.  The  capacity  of  a  tank  is  13  cu.  ft.  576  cu.  in.  How 
many  barrels  of  water  will  it  hold,  if  the  capacity  of  the 
barrel  is  2  cu.  ft.  1152  cu.  in.  ? 

9.  Mnd  the  value  of  15  A.  6S  sq.  rd.  of  land  at  ^225 
an  acre. 

10.  How  many  cannon  balls,  each  weighing  86  lb.  10  oz. 
10|-  dr.,  can  be  formed  from  a  mass  of  iron  weighing  13  cwt.? 

11.  A  merchant  buys  10  barrels  of  wine,  each  containing 
32  gallons,  at  $3.00  a  gallon,  and  sells  it  at  87|-  cents  a 
quart.     How  much  does  he  gain  ? 

12.  A  pipe  fills  a  tank  at  the  rate  of  3  qt.  1  pt.  a  second. 
If  the  capacity  of  the  tank  is  843  gal.  2  qt.,  how  many 
minutes  will  it  take  to  fill  it  ? 

13.  How  many  house-lots,  each  containing  59  sq.  rd. 
26  sq.  yd.  3 J  sq.  ft.,  can  be  made  from  a  piece  of  land  whose 
area  is  114100  sq.  ft.  ? 

14.  If  the  pound  avoirdupois  contains  7000  grains  troy, 
how  many  pennyweights  are  there  in  an  ounce  avoirdupois  ? 

15.  Find  the  sum  of  |  of  dS  5  17s.  8d,  i  of  £  7  5s.  lid., 
and  I  of  £  3  10s.  9d. 

16.  What  is  the  exact  number  of  days  from  Sept.  21, 
1886,  to  April  5,  1892? 

17.  A  man  agreed  to  build  105  rods  of  fence.  On  the 
first  day  he  built  29  rd.  2  yd.,  on  the  second  day  33  rd. 
4  yd.  2  ft.,  and  on  the  third  day  27  rd.  1  yd.  How  much 
remained  to  be  built  at  the  end  of  the  third  day  ? 


122  ARITHMETIC. 

18.  What  is  the  value  of  26  square  chains  of  land,  at 
12  cents  a  square  foot  ? 

19.  Express  2  lb.  11  oz.  17  pwt.  14  gr.  in  apothecaries' 
weight. 

20.  If  the  pound  avoirdupois  contains  7000  grains  troy, 
how  many  drams  are  there  in  an  ounce  troy  ? 

21.  What  is  the  value  of  17160  sq.  ft.  of  land,  at  $800 
an  acre  ? 

22.  If  coal  is  bought  by  the  long  ton  at  $  5.60  a  ton,  and 
sold  by  the  short  ton  at  $  6.25  a  ton,  what  is  the  profit  on 
25  tons  ? 

23.  A  mass  of  granite  weighs  62  lb.  3.8  oz. ;  an  equal 
mass  of  water  weighs  23  lb.  15  oz.  How  many  times  is 
granite  as  heavy  as  water  ? 

24.  If  the  English  sovereign  be  worth  $  4.871  what  is 
the  value  in  cents  of  3s.  Sd.  ? 

25.  If  a  train  travels  at  the  rate  of  36  miles  an  hour, 
what  is  its  rate  in  feet  per  second  ? 

26.  Express  3  lb.  13  oz.  8  dr.  in  apothecaries'  weight. 

27.  If  the  quart  dry  measure  contains  67^  cubic  inches, 
how  many  cubic  feet  are  there  in  a  bushel  ? 

28.  The  cost  of  a  certain  article  in  English  money  is 
3s.  lid.  If  the  sovereign  be  worth  $  4.86,  what  is  the  value 
of  the  article  in  cents  ? 

29.  What  is  the  value  of  a  dozen  silver  spoons,  each 
weighing  3  oz.  13  pwt.  12  gr.,  if  silver  be  worth  f  0.90  an 
ounce  ? 

30.  The  capacity  of  a  tank  is  18  cu.  ft.  516  cu.  in.  If 
the  tank  can  be  emptied  by  a  pipe  in  3  min.  24  sec,  how 
many  cubic  inches  does  it  empty  in  one  second  ? 

31.  Express  8  5  1  3  1  3  5  gr.  in  avoirdupois  weight. 

32.  If  a  man  can  do  a  piece  of  work  in  13  h.  39  min.  12  sec, 
what  part  of  the  work  can  he  do  in  9  h.  6  min.  8  sec.  ? 


DENOMINATE   NUMBERS.  123 

33.  If  the  liquid  quart  contains  57.75  cubic  inches,  how- 
many  cubic  feet  are  there  in  a  hogshead  ? 

34.  Two  places  on  the  equator  are  in  longitude  53°  20'  E., 
and  longitude  71°  55'  W.,  respectively.  Find  the  distance 
between  them  in  miles,  if  the  length  of  a  degree  of  the 
earth's  equator  be  taken  as  69^  miles. 

35.  A  man  who  travelled  from  A  to  B,  found  on  arriving 
there  that  his  watch  was  17  min.  23  sec.  faster  than  the 
local  time.  If  the  longitude  of  A  is  83°  28'  W.,  what  is  the 
longitude  of  B  ? 

36.  If  a  bushel  of  wheat  weighs  60  pounds,  what  is  the 
value  of  a  carload  of  wheat  weighing  9  T.  13  cwt.,  at  93  cents 
a  bushel  ? 

37.  Which  is  the  greater,  f  of  2  gal.  3  qt.  1  pt.,  or  |  of 
3  gal.  2  qt. ;  and  how  much  ? 

38.  If  the  pound  avoirdupois  contains  7000  grains  troy, 
express  the  pound  troy  as  a  decimal  of  a  pound  avoirdupois. 

39.  If  a  furnace  burns  6  T.  6  cwt.  of  coal  from  Nov.  19 
to  Feb.  11,  how  many  pounds  does  it  burn  per  day  ? 

40.  If  a  train  travels  704  inches  a  second,  how  many 
miles  does  it  travel  in  18  min.  54  sec.  ? 

41.  A  vessel,  filled  to  the  brim  with  water,  weighs,  with 
its  contents,  45  pounds.  A  mass  of  copper  is  thrown  into 
it,  displacing  13  lb.  5  oz.  of  water.  If  copper  is  8.8  times 
as  heavy  as  water,  how  much  does  the  vessel  now  weigh  ? 

42.  A  certain  township  contains  4  sq.  mi.  173  A.  120  sq.  rd. 
of  improved,  and  2  sq.  mi.  223  A.  90  sq.  rd.  of  unimproved 
land.     What  part  of  the  township  is  unimproved  ? 

43.  The  local  time  at  two  places  on  the  equator  differs  by 
3  h.  50  min.  24  sec.  What  is  the  distance  in  miles  between 
the  places,  if  the  length  of  a  degree  of  the  equator  is  69^ 
miles  ? 


124  ARITHMETIC. 

44.  If  gold  be  worth  $  20.25  an  ounce  troy,  what  is  the 
value  of  a  mass  of  gold  weighing  28  lb.  11  oz.  avoirdupois  ? 

46.   Express  9  oz.  8  dr.  in  troy  weight. 

46.  If  a  cubic  foot  of  water  weighs  62^  pounds,  and  lead 
is  11.44  times  as  heavy  as  water,  how  many  cubic  inches  will 
there  be  in  a  piece  of  lead  weighing  35  lb.  12  oz.  ? 

47.  The  cost  of  a  certain  article  is  $  2.70.  Find  its  value 
in  English  money,  if  the  sovereign  be  worth  $  4.86. 

48.  A  train  leaves  A  for  B,  26  miles  distant,  travelling 
at  the  rate  of  192  rods  a  minute.  Five  minutes  later,  another 
train  leaves  B  for  A,  travelling  at  the  rate  of  14  yd.  2  ft.  a 
second.     How  many  miles  from  A  will  they  meet  ? 

49.  If  a  gallon  contains  231  cu.  in.,  and  a  bushel  2150.42 
cu.  in.,  express  a  liquid  quart  as  a  decimal  of  a  dry  quart. 

50.  If  a  cubic  foot  of  water  weighs  1000  ounces,  and  iron 
is  7.6  times  as  heavy  as  water,  what  is  the  weight  in  pounds 
of  405  cubic  inches  of  iron  ? 

61.  The  sides  of  a  field  are,  respectively,  3  ch.  17  li.,  2  ch. 
83  li.,  3  ch.  44  li.,  and  2  ch.  68  li.  Find  the  distance  around 
the  field  in  feet. 

62.  A  and  B  buy  a  piece  of  land,  containing  8  A.  155 
sq.  rd.,  for  $7168.  If  A  pays  $2048,  and  B  $5120,  how 
much  land  should  each  receive  ? 

63.  If  the  moon  revolves  around  the  earth  in  27  d.  7  h. 
42  min.,  over  how  many  degrees  of  its  orbit  does  it  pass  in 
1  d.  15  h.  30  min.  ? 

64.  If  a  rail  weighs  63  pounds  to  the  yard,  how  many 
tons  of  rail  will  be  required  to  lay  a  mile  of  single  track 
railway  ? 


THE   METRIC   SYSTEM.  125 


XIII.    THE  METRIC  SYSTEM. 

Note.  The  teacher  may,  at  his  option,  reserve  the  chapter  on  the 
Metric  System  until  the  remaining  chapters  of  the  hook  have  been 
taken,  or  omit  it  entirely. 

175.  In  the  Metric  System  of  measures,  two  consecutive 
denominations  of  the  same  kind  (Art.  158)  are  so  related 
that  the  greater  is  ten  times  the  less. 

Note.  The  Metric  System  is  used  exclusively  in  several  foreign 
countries.  In  the  United  States,  it  is  employed  in  the  sciences,  and 
the  Coast  Survey  ;  it  is  also  used  to  a  certain  extent  in  the  Mint  and 
General  Post  Office. 

176.  Subdivisions  of  the  principal  units  are  expressed  by 
writing  before  the  name  of  the  unit  the  prefixes  : 

Milli-,  to  denote  one-thousandth  of  the  unit ; 
Centi-,  to  denote  one-hundredth  of  the  unit ; 
Deci-f   to  denote  one-tenth  of  the  unit. 

Multiples  of  the  principal  units  are  expressed  by  writing 
before  the  name  of  the  unit  the  prefixes : 

Dekor,    to  denote  ten  times  the  unit ; 
Hekto-,  to  denote  one  hundred  times  the  unit; 
Kilo-f     to  denote  one  thousand  times  the  unit ; 
Myriorj  to  denote  ten  thousand  times  the  unit. 

Only  a  few  of  the  denominations  are  much  used ;  these 
will  be  indicated  in  the  following  tables  by  printing  them 
in  larger  type. 

177.  Measures  of  Length. 

The  principal  unit  of  length  is  the  Meter ;  it  is  approx- 
imately equal  to  one  ten-millionth  part  of  the  distance, 
measured  on  a  meridian,  from  the  equator  to  the  pole. 


126  ARITHMETIC. 

TABLE. 

10  millimeters  (""")  =  1  centimeter.     ("•") 

10  centimeters  =  1  decimeter.     (*^'") 

10  decimeters  =  1  meter.     (") 

10  meters  =  1  dekameter.     (^™) 

10  dekameters  =  1  hektometer.     (^™) 

10  hektometers  =  1  kilometer.     (^'") 

10  kilometers  =  1  myriameter.     (^'") 

EQUIVALENTS. 

1  centimeter  =  .3937  in.  1  inch  =  2.54*^. 

1  meter  =  39.37  in.  1  foot  =  30.48««'. 

1  meter  =  1.0936  yd.  1  rod  =  5.029°». 

1  kilometer   =  .6214  mi.  1  mile  =  1.6093^"^. 

Note  1.  The  millimeter  and  centimeter  are  used  for  very  small 
measurements  in  the  arts  and  sciences  ;  the  meter,  in  measuring  cloth, 
short  distances,  etc.  ;  the  kilometer,  in  measuring  long  distances. 

Note  2.    A  kilometer  is  approximately  f  of  a  mile. 

178.  Measures  of  Area. 

The  principal  unit  of  area  is  the  Square  Meter ;  that  is, 
the  area  of  a  square  whose  side  is  one  meter. 

TABLE. 

100  square  millimeters  (^i^n™)  =  l  square  centimeter.    ("!'='") 
100  square  centimeters  =  1  square  decimeter,     (sq**'") 

100  square  decimeters  =  1  square  meter.     C^™) 

100  square  meters  =  1  square  dekameter.  ("iDm^ 

100  square  dekameters  =  1  square  hektometer.  {'"^  ^'") 

100  square  hektometers  =  1  square  kilometer.     ("^  ^'") 


THE  METRIC   SYSTEM.  127 

The  following  terms  are  also  used : 

A  centar  (•"*)  =  a  square  meter. 

An  ar  (*)        =  a  square  dekameter,  or  100  centars. 

A  hektar  (^*)  =  a  square  hektometer,  or  100  ars. 

EQUIVALENTS. 
;[sqcm  ^  j^55  gq^  ijj  I  gq  ^^   _  e.4528qcm^ 

1^""   =  1.196  sq.  yd.  1  sq.  yd.  =  .SSGl""!™ 

^sqKm  ^  335;!^  g^^  ^^  ^  ^^  j,^   ^  .2529^ 

1^       =  3.954  sq.rd.  1  sq.  mi.  =  2.59^^^'". 

l""^     =  2.471  A.  1  A.         =  .4047H^ 

Note.  The  square  meter  is  used  in  measuring  ordinary  surfaces ; 
the  square  kilometer,  in  measuring  the  areas  of  countries  ;  the  ar  and 
hektar,  in  measuring  land. 

179.  Measures  of  Volume. 

The  principal  unit  of  volume  is  the  Cubic  Meter ;  that  is, 
the  volume  of  a  cube  whose  edge  is  one  meter. 

TABLE. 

1000  cubic  millimeters  (c"""™)  =  1  cubic  centimeter.      C^^'^") 
1000  cubic  centimeters  =  1  cubic  decimeter.     (•="  '^™) 

1000  cubic  decimeters  =  1  cubic  meter.     (^" '") 

The  following  terms  are  also  used: 

A  decister  (^^)  =  .1  cubic  meter. 

A  ster  C^)  =  1  cubic  meter,  or  10  decisters. 

EQUIVALENTS. 

jcucm  ^  .06103  cu.  in.  1  cu.  in.  =  16.387^"'='". 

j^cum  ^  j^  308  cu.  yd.  1  cu.  yd.  =  .7645'="'". 

l«t      =  .2759  cord.  1  cord      =  3.624«^ 

Note.  The  cubic  meter  is  used  in  measuring  ordinary  solids  ;  the 
ster,  in  measuring  wood. 


128  ARITHMETIC. 

180.  Measures  of  Capacity. 

The  principal  unit  of  capacity  is  the  Liter,  which  is  equal 
to  a  cubic  decimeter. 

TABLE. 


10  milliliters  {""')  =  1  centiliter. 

n 

10  centiliters 

=  1  deciliter. 

n 

10  deciliters 

=  1  Uter.     Q) 

10  liters 

=  1  dekaliter. 

n 

10  dekaliters 

=  1  hektoUter. 

n 

10  hektoliters 

=  1  kiloliter. 

n 

EQUIVALENTS. 

1  liter  =  61.022  cu.  in.  1  hektoliter  =  2.837  bu. 

1  liter  =  1.0567  liq.  qt.  1  liquid  qt.  =  .9463^ 

1  liter  =  .9081  dry  qt.  1  dry  qt.       =  1.101\ 

1  hektoliter  =  3.531  cu.  ft.  1  gallon        =  3.785^ 

1  hektoliter  =  26.417  gal.  1  bushel       =  .3524^1. 

Note.  The  liter  is  used  in  measuring  liquids  and  small  fruit ;  the 
hektoliter,  in  measuring  grain,  vegetables,  and  liquids  in  casks. 

181.   Measures  of  Weight. 

The  principal  unit  of  weight  is  the  Gram;  that  is,  the 
weight  of  a  cubic  centimeter  (milliliter)  of  distilled  water 
at  its  greatest  density  (39.2°  Fahrenheit). 

TABLE. 

10  milligrams  ('"«)  =  1  centigram.     C^^) 
10  centigrams  =  1  decigram.     (*^^) 

10  decigrams  =  1  gram.     (^) 

10  grams  =  1  dekagram.     {^^) 

10  dekagrams  =  1  hektogram.     (^^) 

10  hektograms         =  1  kilogram.     {^^) 


THE   METRIC   SYSTEM.  129 

10  kilograms      =  1  myriagram.     (**^) 
10  myriagrams  =  1  quintal.     (^) 
10  quintals        =  1  metric  ton.     C^) 

EQUIVALENTS. 

1  gram  =  15.432  gr.  1  oz.  troy  =  31.1035k. 

1  gram  =  .03527  oz.  av.  1  oz.  av.    =  28.35^. 

1  kilogram    =  2.2046  lb.  av.  1  lb.  av.    =  .4536^«. 

1  metric  ton  =  1.1023  T.  1  grain      =  .0648^. 

Note  1.    The  quintal  is  very  seldom  used. 

Note  2.  The  gram  is  used  in  weighing  letters,  precious  metals, 
and  jewels,  and  in  mixing  medical  prescriptions  ;  the  kilogram,  in 
weighing  ordinary  articles  ;  the  metric  ton,  in  weighing  heavy  articles. 

Note  3.  A  kilogram  is  the  weight  of  a  cubic  decimeter  (liter), 
and  a  metric  ton  of  a  cubic  meter  (kiloliter),  of  distilled  water  at  its 
greatest  density. 

Note  4.  Of  the  United  States  coins,  the  nickel  five-cent  piece 
weighs  5  grams,  and  two  silver  half-dollars  weigh  25  grams. 

Note  5.    A  kilogram  is  approximately  2^  pounds  avoirdupois. 

EXAMPLES. 
182.   1.  How  many  centimeters  are  there  in  a  dekameter? 

2.  How  many  milligrams  are  there  in  a  hektogram  ? 

3.  How  many  square  centimeters  are  there  in  a  square 
meter  ?, 

4.  How  many  square  meters  are  there  in  a  hektar  ? 

5.  How  many  cubic  millimeters  are  there  in  a  cubic 
decimeter  ? 

6.  How  many  deciliters  are  there  in  a  cubic  meter  ? 

7.  How  many  hektograms  are  there  in  a  metric  ton  ? 

8.  How  many  millimeters  are  there  in  a  kilometer  ? 

9.  How  many  square  decimeters  are  there  in  a  square 
hektometer  ? 

10.  How  many  cubic  millimeters  are  there  in  a  ster  ? 

11.  How  many  milliliters  are  there  in  a  dekaliter  ? 


130  ARITHMETIC. 


METRIC  NUMBERS. 


183.  A  Metric  Number  is  a  denominate  number  (Art. 
160)  expressed  in  denominations  of  the  metric  system. 

A  metric  number  is  usually  expressed  in  terms  of  a  single 
denomination  of  the  same  kind. 

Thus,  7^"  SH'"  9°™  l'"  2*^™  S'^'"  may  be  expressed  in  the  forms 
73.9128«™,  73912.8'*-  etc. 

To  read  a  metric  number  when  expressed  in  terms  of  a 
single  denomination,  the  name  of  the  denomination  may  be 
given  to  the  number  to  the  left  of  the  decimal  point,  and 
the  name  of  the  smallest  denomination  to  the  portion  to  the 
right  of  the  decimal  point. 

Thus,  73.9128^'"  maybe  read  either  "73,  and  9128  ten- 
thousandths  hektometers,"  or  "73  hektometers,  and  9128 
centimeters." 

184.  Reduction  of  Metric  Numbers. 

A  metric  number  may  be  expressed  as  a  decimal  of  the 
next  lower  denomination  by  moving  the  decimal  point  one 
place  to  the  right ;  as  a  decimal  of  the  next  lower  denomi- 
nation but  one  by  moving  the  decimal  point  two  places  to 
the  right;  and  so  on.     (Compare  Art.  144.) 

Thus,  73.9128H'"  =  739.128°"^  =  7391.28'",  etc. 

A  metric  number  may  be  expressed  as  a  decimal  of  the 
next  higher  denomination  by  moving  the  decimal  point  one 
place  to  the  left ;  as  a  decimal  of  the  next  higher  denomi- 
nation but  one  by  moving  the  decimal  point  two  places  to 
the  left;  and  so  on.     (Compare  Art.  145.) 

Thus,  5.301'"g  =  .5301'^s  =  .05301^^,  etc. 

The  above  rules  do  not  apply  to  the  table  of  measures 
of  area  and  volume  given  in  Arts.  178  and  179;  in  these 
cases,  the  decimal  point  must  be  moved  two  and  three  places, 
respectively. 

Thus,  85.64''i»™  =  8564"^'"  =  856400"^^°^  etc. ; 

JgQcu  mm  ^  Jgcu  cm     ^  .00078""  '*'",   CtC. 


THE  METRIC   SYSTEM.  131 


EXAMPLES. 


1.  Express  4.721^  as  a  decimal  of  a  centigram. 

2.  Express  .07319^"  as  a  decimal  of  a  decimeter. 

3.  Express  8943*"^  as  a  decimal  of  a  kiloliter. 

4.  Express  32.75'"  as  a  decimal  of  a  hektometer,  of  a 
dekameter,  and  of  a  millimeter. 

5.  Express  .05487"^  ^"^  as  a  decimal  of  a  square  meter,  of 
a  square  kilometer,  and  of  a  square  centimeter. 

6.  Express  132*^^  as  a  decimal  of  a  hektoliter,  of  a  liter, 
and  of  a  centiliter. 

7.  Express  .00651^^  in  ars,  and  in  centars. 

8.  Express  .09078^^  in  metric  tons,  in  hektograms,  and 
in  decigrams. 

9.  Express  .732°™  in  kilometers,  in  meters,  and  in  centi- 
meters. 

10.  Express  4615*^"'"'"  as  a  decimal  of  a  cubic  decimeter, 
and  of  a  cubic  centimeter. 

11.  Express  .002505^^  in  dekaliters,  in  deciliters,  and  in 
milliliters. 

12.  How  many  decisters  are  there  in  1834*="  ^"^  ? 

13.  Express  64.94*=^  as  a  decimal  of  a  kilogram,  of  a  deka- 
gram, and  of  a  milligram. 

14.  What  is  the  weight  in  hektograms  of  a  cubic  centi- 
meter of  distilled  water  at  its  greatest  density  ? 

15.  What  is  the  weight  in  decigrams  of  a  centiliter  of 
distilled  water  at  its  greatest  density  ? 

16.  Express  21.2^  ^'^  in  centars. 

17.  Express  .908'="*='°  in  centiliters. 

18.  Find  the  weight  in  myriagrams  of  14.04''""'  of  water. 

19.  Express  10.7^*  in  square  kilometers. 

20.  How  many  deciliters  of  water  will  weigh  .000686*=^  ? 


132  ARITHMETIC. 

21.  Express  7.13^'  in  cubic  dekameters. 

22.  Express  863'^"''™  in  decisters. 

23.  Express  .0359*="^'"  in  hektoliters. 

24.  How  many  dekaliters  of  water  will  weigh  5.257''  ? 

25.  Express  .48352*=^  of  water  in  cubic  millimeters,  and 
find  its  weight  in  centigrams. 

185.  Addition,  Subtraction,  Multiplication,  and  Division 
of  Metric  Numbers. 

If  two  metric  numbers  be  expressed  as  decimals  of  the 
same  denomination,  they  may  be  added  or  subtracted  by  the 
methods  of  Arts.  119  or  120;  the  result  being  a  decimal  of 
the  same  denomination. 

They  may  also  be  divided  by  the  method  of  Art.  127. 

Again,  a  metric  number  may  be  multiplied  or  divided  by 
the  methods  of  Arts.  121  or  127 ;  the  result  being  a  decimal 
of  the  same  denomination. 

1.   Add  3.561^^  .0903^\  47.07^,  and  862.8'=l 

Expressing  each  metric  number  as  the  decimal  of  a  liter,  we  have 

356.1^ 

.903 
47.07 

8.628 


412.701^  Ans. 

2.   Divide  3.71622^2  by  72.3*=«. 

Expressing  3.71622Dg  as  the  decimal  of  a  centigram,  we  have 
72.3)3716.22(51.4,  Ans. 
3615 

1012 
723 

2892 
2892 


THE  METRIC   SYSTEM.  133 

EXAMPLES. 

3.  Add  5621^',  .01906«i,  .3027^',  and  4.249'^^ 

4.  Add  .153^8,  78.26'"«,  8.53^^,  and  .006097^^. 

5.  Add  47.6™,  83.057"'",  .04829"'",  and  9720.4^™. 

6.  Add  S-IU^^"-",  30135«'i'='",  and  .00598'^'". 

7.  Subtract  89.7°«  from  2.8164^^ 

8.  Subtract  2.482'"'"  from  .50168"'. 

9.  Subtract  8.5819'="'^"  from  9516.27*^'^'=™. 

10.  Subtract  60.433^'  from  4.307«^ 

11.  Multiply  5.03"™  by  2.7,  and  express  the  result  as  a 
decimal  of  a  hektometer. 

12.  Multiply  .8259^^™  by  30.8,  and  express  the  result  as 
a  decimal  of  a  square  meter. 

13.  Multiply  43.7™^  by  .519,  and  express  the  result  as  a 
decimal  of  a  centigram. 

14.  Multiply  .002846"^  by  .0733,  and  express  the  result  as 
a  decimal  of  a  deciliter. 

15.  Divide  93.98™^  by  25.4^. 

16.  Divide  .0737505^^  by  40.5,  and  express  the  result  as 
a  decimal  of  a  dekagram. 

17.  Divide  4.686"™  by  825'". 

18.  Divide  .0602784<="  *^™  by  .644,  and  express  the  result  as 
a  decimal  of  a  cubic  millimeter. 

186.  To  express  a  Metric  Number  in  ordinary  Units,  or  an 
ordinary  Denominate  Number  in  Metric  Units. 

Problems  of  this  kind  may  be  solved  by  means  of  the 
tables  of  equivalents  given  in  Arts.  177  to  181. 

1.   Express  3"™  in  feet. 

We  have,  3Dm  =  so™. 

But  by  Art,  J77,  a  meter  is  equivalent  to  1.0936  yd. 

Then,  SOm  =  30  x  1.0936  yd. 

=  32.808  yd.  =  98.424  ft.,  Ans. 


134  ARITHMETIC. 

2.   Express  135  lb.  10  oz.  8  dr.  in  hektograms. 
We  have,  135  lb.  10  oz.  8  dr.  =2170.5  oz. 

But  by  Art.  181,  an  ounce  avoirdupois  =  28.35g. 
Then,  2170.5  oz.  av.  =  2170.5  x  28.358 

=  61533.675K. 
Whence,  135  lb.  10  oz.  8  dr.  =  615.33675^??,  Ans. 

EXAMPLES. 

3.  How  many  decimeters  are  there  in  a  yard  ? 

4.  How  many  dekaliters  are  there  in  a  peck  ? 

5.  How  many  hundredweights  are  there  in  a  myriagram  ? 

6.  How  many  cubic  yards  are  there  in  a  decister  ? 

7.  How  many  square  decimeters  are  there  in  a  square 
foot? 

8.  How  many  cubic  feet  are  there  in  a  cubic  decimeter  ? 

9.  How  many  gills  are  there  in  a  centiliter  ? 

10.  How  many  square  feet  are  there  in  a  centar  ? 

11.  How  many  dekagrams  are  there  in  a  dram  ? 

12.  How  many  feet  are  there  in  a  dekameter  ? 

13.  How  many  deciliters  are  there  in  a  liquid  pint  ? 

14.  How  many  hektograms  are  there  in  a  pound  troy  ? 

15.  How  many  cubic  yards  are  there  in  a  kiloliter  ? 

16.  Express  31^°*  in  rods. 

17.  Express  4.7  pwt.  in  centigrams. 

18.  Express  75.1*^^  in  cubic  inches. 

19.  Express  .642  sq.  mi.  in  hektars. 

20.  Express  29  cu.  ft.  in  hektoliters. 

21.  Express  .0083^  in  hundredweights. 

22.  Express  .93^«  in  drams. 

23.  Express  5  cu.  ft.  in  cubic  decimeters. 

24.  Express  288  cu.  ft.  in  decisters. 


THE  METRIC   SYSTEM.  135 

25.  Express  .0153  pt.  dry  measure  in  centiliters. 

26.  Express  .584'*^'"  in  square  rods. 

27.  Express  10  bu.  2  pk.  5  qt.  in  dekaliters. 

28.  Express  5  yd.  1  ft.  11  in.  in  dekameters. 

29.  Express  8  sq.  yd.  5  sq.  ft.  112  sq.  in.  in  square  meters. 

30.  Express  20  cu.  ft.  1440  cu.  in.  in  sters. 

31.  Express  2  lb.  13  oz.  4  dr.  in  centigrams. 

32.  Express  3  pk.  7  qt.  1  pt.  in. cubic  decimeters. 

33.  Express  4  gal.  1  qt.  0  pt.  2  gi.  in  deciliters. 

34.  Express  12  cd.  96  cu.  ft.  in  cubic  dekameters. 

35.  Express  8  lb.  3  oz.  10  pwt.  in  kilograms. 

36.  Express  5  A.  68  sq.  rd.  13  sq.  yd.  in  hektars. 

37.  Express  1  mi.  181  rd.  3  yd.  2  ft.  6  in.  in  meters. 

MISCELLANEOUS  PROBLEMS. 
187.    1.    How  many  dekameters  are  there  in  a  chain  ? 

2.  How  many  square  rods  are  there  in  a  square  hekto- 
meter  ? 

3.  If  a  ream  of  paper  is  .7872^^™  in  thickness,  what  is 
the  thickness  in  millimeters  of  a  single  sheet  ? 

4.  How  much  is  gained  by  buying  3.64^^  of  land  at 
$  1025  a  hektar,  and  selling  it  at  13^  cents  a  square  meter  ? 

5.  How  man}^  dry  pints  are  there  in  a  deciliter  ? 

6.  How  many  dekagrams  are  there  in  a  scruple  ? 

7.  If  a  horse  eats  2^  of  oats  in  a  week,  how  many  weeks 
will  3  bushels  last  him  ? 

8.  A  merchant  buys  ISO""  of  silk  for  $247.70,  and  sells  it 
at  $  1.75  a  yard.     How  much  does  he  gain  ? 

9.  How  much  do  I  lose  by  buying  49.763^^  of  grain,  at 
$  2.88  a  hektoliter,  and  selling  it  at  2.39  cents  a  cubic  deci- 
meter ? 


136  ARITHMETIC. 

10.  How  many  pounds,  apothecaries'  weight,  are  there  in 
a  kilogram  ? 

11.  If  sulphuric  acid  is  1.84  times  as  heavy  as  water 
what  is  the  weight  in  dekagrams  of  26^  of  the  acid  ? 

12.  If  alcohol  is  .791  times  as  heavy  as  water,  how  many 
dekaliters  of  alcohol  will  it  take  to  weigh  3061.17^8  ? 

13.  How  many  deciliters  are  there  in  a  fluid  dram  ? 

14.  How  many  ars  are  there  in  a  square  chain  ? 

15.  How  many  links  are  there  in  a  decimeter  ? 

16.  A  merchant  buys  300^  of  wine  at  $0.83  a  liter.  At 
what  price  per  gallon  must  he  sell  it  to  gain  $  50  ? 

17.  A  tank  contains  4.2966'^  of  water.  How  long  will  it 
take  to  empty  it  by  a  pipe  through  which  pass  1.54^^  a 
second  ? 

18.  A  block  of  stone  weighs  15  T.  4  cwt.  20  lb.  How 
many  cubic  meters  does  it  contain,  if  it  is  2.6  times  as  heavy 
as  an  equal  bulk  of  water  ? 

19.  How  many  centiliters  are  there  in  a  gill  ? 

20.  A  merchant  buys  silk  at  the  rate  of  $  2.35  a  yard.  At 
what  price  per  meter  must  he  sell  it,  so  as  to  neither  gain 
nor  lose  by  the  transaction  ? 

21.  The  scale  of  a  map  is  36'"'"  to  a  kilometer.  If  the 
measured  distance  between  two  places  on  the  map  is 
48.276'^'",  what  is  the  actual  distance  between  them  in 
hektometers  ? 

22.  How  many  decigrams  are  there  in  a  pennyweight  ? 

23.  How  many  drams  are  there  in  a  dekagram  ? 

24.  What  is  the  weight  in  kilograms  of  a  cubic  foot  of 
water  ? 

25.  A  tank,  containing  738.4^^  of  water,  has  two  taps. 
One  fills  it  at  the  rate  of  .017"""  a  minute,  and  the  other 
empties  the  contents  at  the  rate  of  5*^^  a  second.  How  long 
will  it  take  to  empty  the  tank  if  both  pipes  are  opened  ? 


THE  METRIC   SYSTEM  _  137 

26.  Find  the  weight  of  a  pint  of  water  in  dekagrams. 

27.  If  a  train  runs  at  the  rate  of  51^""  an  hour,  what  is 
its  rate  in  rods  a  minute  ? 

28.  How  many  minims  are  there  in  a  centiliter  ? 

29.  If  it  costs  $  5  to  travel  384^"^  by  rail,  what  is  the  rate 
of  fare  in  cents  per  mile  ? 

30.  If  a  man  walks  at  the  rate  of  3.6  miles  an  hour,  how 
many  minutes  will  it  take  him  to  walk  a  kilometer  ? 

31.  How  many  cubic  inches  of  water  weigh  a  hektogram  ? 

32.  How  many  gallons  of  water  weigh  a  metric  ton  ? 

33.  If  a  horse  travels  at  the  rate  of  11.2^™  an  hour,  how 
many  miles  can  he  travel  in  4  h.  39  min.  ? 

34.  A  tank  can  be  filled  with  water  in  2  h.  47  min.  by  a 
tap  through  which  pass  75^^  a  minute.  If  the  tank  is  filled 
with  kerosene,  how  much  are  the  contents  worth  at  28  cents 
a  dekaliter  ? 

35.  If  a  cubic  yard  of  masonry  weighs  3500  lb.,  what  is 
the  weight  of  a  cubic  decimeter  in  kilograms  ? 

36.  If  a  cubic  inch  of  bronze  weighs  .3032  lb.,  how  many 
cubic  centimeters  will  it  take  to  weigh  a  hektogram  ? 

37.  If  a  cubic  inch  of  mercury  weighs  7.88  oz.  avoirdupois, 
how  much  does  a  liter  weigh  in  kilograms  ? 

38.  If  sea-water  is  1.026  times  as  heavy  as  fresh  water, 
how  many  cubic  yards  of  sea-water  will  it  take  to  weigh  a 
metric  ton  ? 

39.  If  a  rail  weighs  69  pounds  to  the  yard,  how  many 
kilograms  does  it  weigh  to  the  meter  ? 

40.  Two  places  on  the  equator  are  in  Ion.  42°  37'  57"  E., 
and  Ion.  78°  34'  3''  W.,  respectively.  Find  the  distance 
between  them  in  kilometers,  if  a  degree  of  the  equator  be 
taken  as  69^  miles. 


138  ARITHMETIC. 


XIV.    INVOLUTION  AND  EVOLUTION. 

188.  Involution  is  the  process  of  raising  a  number  to 
any  required  power  (Art.  55). 

This  may  be  effected  by  taking  the  given  number  as  a 
factor  as  many  times  as  there  are  units  in  the  exponent  of 
the  required  power  (Art.  55). 

1.  Find  the  value  of  (f)^ 

We  have,         (|)'  =  |x|xfxf  =  ^\\,  Ans. 

The  following  rule  is  evident  from  the  above : 

To  raise  a  fraction  to  any  power,  raise  both  numerator  and 

denominator  to  the  required  power,  and  divide  the  first  result 

by  the  second. 

EXAMPLES. 
Find  the  values  of  the  following : 

2.  87^.        4.   1033.        6.   12^        8.    (^y.        10.    {^y. 

3.  613.        5_   25^  7.    {\y.     9.    (3f)l        11.    (|)^ 

189.  If  one  number  is  the  square  of  another,  the  second 
number  is  said  to  be  the  Square  Root  of  the  first. 

Thus,  since  9  =  3^,  3  is  the  square  root  of  9. 

190.  If  one  number  is  the  cube  of  another,  the  second 
number  is  said  to  be  the  Cube  Root  of  the  first. 

Thus,  since  8  =  2^,  2  is  the  cube  root  of  8. 

191.  In  like  manner,  if  one  number  is  the  fourth  power 
of  another,  the  second  number  is  said  to  be  the  Fourth 
Root  of  the  first ;  and  so  on. 

192.  Evolution  is  the  process  of  finding  any  required  root 
of  a  number. 

193.  The  Radical  Sign,  V  ,  when  written  over  a  number, 
indicates  some  root  of  the  number. 


INVOLUTION  AND   EVOLUTION.  139 

Thus,       V25  indicates  the  square  root  of  25 ; 
V  27  indicates  the  cube  root  of  27 ; 
V81  indicates  the  fourth  root  of  81 ;  etc. 

The  Index  of  a  root  is  the  number  written  over  the  radical 
sign  to  indicate  what  root  of  the  number  is  taken. 
Thus,  in  V27,  the  index  is  3. 

194.  If  one  number  is  a  power  of  another,  the  first  num- 
ber is  said  to  be  a  perfect  power  of  the  degree  denoted  by  the 
exponent  of  the  power. 

A  perfect  power  of  the  second  degree  is  called  a  perfect 
square,  and  a  perfect  power  of  the  third  degree  is  called  a 
perfect  cube. 

Thus,  since  7^  =  49,  49  is  a  perfect  square. 

195.  It  is  evident  from  Ex.  1,  Art.  188,  that  to  find  any 
root  of  a  fraction,  each  of  whose  terms  is  a  perfect  power  of 
the  degree  denoted  by  the  index  of  the  required  root,  we 
extract  the  required  root  of  both  numerator  and  denominator, 
and  divide  the  first  result  by  the  second. 

196.  1.   Extract  the  cube  root  of  343. 

We  find  a  number  whose  cube  is  equal  to  343. 
The  number  is  7 ;  hence,  V343  =  7,  Ans. 

2.   Extract  the  fourth  root  of  Hf. 

^     .   .    .^.    4/256      ■\/2E6      4    , 


"^^^^•^^"'^625     ^e25     5' 

Lfl>0. 

EXAMPLES. 

Find  the  values  of  the  following  : 

'3.    V121.          5.    v'1296.          7. 

VW- 

4.    ^125.          6.    ^1024.          8. 

m 

9.  </u. 


140  ARITHMETIC. 

SQUARE    ROOT. 

197.  Let  it  be  required  to  find  the  square  of  74. 
We  have,  74  =  70  +  4. 

To  multiply  70  +  4  by  itself,  we  multiply  it  by  70,  and 
then  by  4,  and  add  the  second  result  to  the  first. 

70  X  (70  +  4)  =  702-f         70x4 
4  X  (70  +  4)=  70x4  +  4=* 

Whence,  74^  =  70^  +  2  x  70  x  4  +  41 

That  is,  the  square  of  any  number  of  two  figures  is  equal  to 
the  square  of  the  tens,  plus  twice  the  product  of  the  tens  by  the 
units,  plus  the  square  of  the  units. 

198.  It  follows  from  Art.  197  that 

742-702  =  2x70x4  +  42 
=  (2  X  70  +  4)  X  4. 

That  is,  if  from  the  square  of  any  number  of  two  figures  the 
square  of  the  tens  be  subtracted,  the  remainder  is  equal  to  twice 
the  tens,  plus  the  units,  multiplied  by  the  units. 

199.  We  have  1'  =  1,  9^  =  81,  10^  =  100,  99^  =  9801,  etc. 
That  is,  the  square  of  any  number  of  one  figure  contains 

either  one  or  two  figures ;  the  square  of  any  number  of  two 
figures  contains  either  three  or  four  figures ;  etc. 

Hence,  if  a  point  be  placed  over  every  second  figure  of  any 
number,  beginning  at  the  units'  place,  the  number  of  points 
shows  the  number  of  figures  in  its  square  root. 

200.  1.   Find  the  square  root  of  6889. 

6889  I  80  +  3  =  83,  Ans. 
802  ^  (3400 


Trial  divisor,  2  x  80  =  160 

3 
Complete  divisor,  163 


489 
489 


INVOLUTION  AND  EVOLUTION.  141 

Pointing  the  number  by  the  rule  of  Art.  199,  we  find  that  there  are 
two  figures  in  its  square  root. 

The  greatest  perfect  square  in  68  is  64,  which  is  the  square  of  8 ; 
then  8  is  the  tens'  figure  of  the  root. 

Subtracting  the  square  of  80,  or  6400,  from  6889,  the  remainder  is 
489. 

By  Art.  198,  this  is  equal  to  2  x  80,  plus  the  units'  figure  of  the  root, 
multiplied  by  the  units'  figure  of  the  root. 

That  is,  489  is  equal  to  160,  plus  the  units'  figure,  multiplied  by  the 
units'  figure. 

Then,  to  obtain  the  units'  figure,  we  must  divide  489  by  160  plus  the 
units'  figure. 

Now  we  can  find  an  approximate  value  of  the  units'  figure  by  divid- 
ing 489  by  160. 

The  quotient  is  3+  ;  we  then  infer  that  the  units'  figure  is  3. 

Adding  3  to  the  trial-divisor^  160,  we  obtain  the  complete  divisor^ 
168  ;  multiplying  this  by  3,  the  product  is  489,  which,  subtracted  from 
489,  leaves  no  remainder. 

Then  the  required  square  root  is  83. 

Omitting  the  ciphers  for  the  sake  of  brevity,  and  condens- 
ing the  process,  it  will  stand  as  follows : 

6889  I  83 
64 


163 


489 
489 


From  the  above  example,  we  derive  the  following  rule  : 

Separate  the  number  into  periods  by  pointing  every  second 
figure  beginning  with  the  units^  place. 

Find  the  greatest  square  in  the  left  hand  period,  and  write 
its  square  root  as  the  first  figure  of  the  root;  subtract  the  square 
of  the  first  root-figure  from  the  first  period,  and  to  the  result 
annex  the  next  period.     ~ 

Divide  this  remainder,  omitting  the  last  figure,  by  twice  the 
part  of  the  root  already  obtained,  and  annex  the  quotient  to 
the  root,  and  also  to  the  trial-divisor. 

Midtiply  the  complete  divisor  by  the  rootfigure  last  obtained, 
and  subtract  the  product  from  the  remainder. 


45 

227 
225 

500 

4 

20016 
20016 

142  ARITHMETIC. 

If  other  periods  remain,  we  may  regard  the  part  of  the 
root  already  found  as  tens  with  respect  to  the  next  root- 
figure,  and  proceed  as  before ;  doubling  the  part  of  the  root 
already  found  for  the  next  trial-divisor. 

If  any  root-figure  is  0,  annex  0  to  the  trial-divisor,  and 
annex  to  the  remainder  the  next  period.     (Compare  Ex.  2.) 

2.   Find  the  square  root  of  6270016. 

•    •    •    •,  In  this  case,  the  second  trial-divisor 

6270016  1  2504,  Ans.     j^  50^  and  the  second  remainder  is 
4  200. 

Since  50  is  not  contained  in  20,  the 
third  root-figure  is  0. 

We  then  annex  0  to  the  trial-divisor 
60,  giving  500,  and  annex  to  the  re- 
mainder the  next  period,  16. 

If,  on  multiplying  any  complete  divisor  by  the  root-figure 
last  obtained,  the  product  is  greater  than  the  remainder, 
the  figure  of  the  root  last  obtained  is  too  great,  and  one  less 
must  be  substituted  for  it. 

201.  Square  Boot  of  Decimals. 

We  have  1.5^  =  2.25,  .61^  =  .3721,  etc. 

That  is,  the  square  of  a  decimal  of  one  place  is  a  decimal 
of  two  places ;  the  square  of  a  decimal  of  two  places  is  a 
decimal  of  four  places  ;  etc. 

Hence,  if  a  point  be  placed  over  every  second  figure  of  any 
decimal,  beginning  with  the  units'  place,  and  extending  in 
either  direction,  the  number  of  points  to  the  left  of  the  decimal 
point  shows  the  number  of  places  in  the  integral,  and  the 
number  of  points  to  the  right,  the  number  of  places  in  the  deci- 
mal, portion  of  the  square  root. 

202.  The  rule  of  Art.  200  may  be  used  to  find  the  square 
root  of  a  decimal,  provided  that  the  decimal  point  is  inserted 
in  its  proper  position  in  the  root. 


INVOLUTION   AND  EVOLUTION. 

Example.     Eind  the  square  root  of  14.3641. 
14.364i  I  3.79,  Ans. 


143 


67 

536 
469 

749 

6741 
6741 

EXAMPLES. 
203.   Find  the  square  roots  of  the  following : 


1. 

6084. 

7. 

.247009. 

13. 

81U40.64. 

2. 

.9409. 

8. 

.00521284. 

14. 

33443089. 

3. 

457.96. 

9. 

33.1776. 

16. 

58.38488:j. 
.0022448644. 

4. 

273529. 

10. 

.01170724. 

16. 

5. 

.081796. 

11. 

446.0544. 

17. 

738482.4225. 

6. 

6544.81. 

12. 

.68112009. 

18. 

.9733203649. 

204.  Square  Root  of  an  Imperfect  Square. 

If  there  is  a  final  remainder,  the  number  has  no  exact 
square  root ;  but  we  may  continue  the  process  by  annexing 
periods  of  ciphers,  and  obtain  an  approximate  value  of  the 
root,  correct  to  any  desired  number  of  decimal  places. 

1.   Mnd  the  square  root  of  12  to  four  decimal  places. 

12.06060606  I  3.4641+,  Ans. 
9 


64 

300 
256 

686 

4400 
4116 

6924 

28400 
27696 

692 

81 

70400 
69281 

144  ARITHMETIC. 

EXAMPLES. 

Extract  the  square  roots  of  the  following  to  four  places 
of  decimals  : 

2.  7.  4.   17.3.  6.   .1.  8.   375.8329. 

3.  31.  6.   .08.  7.  2.08627.  9.   684.625. 

To  find  the  approximate  square  root  of  a  fraction  whose 
denominator  is,  and  whose  numerator  is  not,  a  perfect 
square,  we  may  divide  the  approximate  square  root  of  the 
numerator  by  the  square  root  of  the  denominator. 

If  the  denominator  is  not  a  perfect  square,  the  fraction 
should  be  reduced  to  an  equivalent  fraction  whose  denomi- 
nator is  a  perfect  square. 

10.  Find  the  square  root  of  |  to  four  places  of  decimals. 

J3^     /I^V6^2i494i^,ei23+,^... 
A/8       \16       4  4  ' 

Extract  the  square  roots  of  the  following  to  four  places  of 
decimals : 

11.  J^.         13.   i^.         15.  A-        17.   A-         19-  U- 

12.  A.         14.   i.  16.   A-        18.  M-         20.  ,%. 


CUBE    ROOT. 

205.   Let  it  be  required  to  find  the  cube  of  74. 
By  Art.  197,  74^  =  70^  +  2  x  70  x  4  +  41 
To  multiply  the  above  result  by  74,  we  multiply  it  by  70, 
and  then  by  4,  and  add  the  second  result  to  the  first. 

70  X  (70^+2  X  70  X 4+42)  =703+2  ^  702 x 4+       70  x 4^ 
4  X  (702+2x70x4+42)=  702x4+2x70x42+4^ 

Whence,  743=70''+3  x  702  x  4+3  x  70  x  42.^43  • 

That  is,  the  cube  of  any  number  of  two  figures  is  equal  to 
the  cube  of  the  tens,  plus  three  times  the  square  of  the  tens 
times  the  units,  plus  three  times  the  tens  times  the  square  of  the 
units,  plus  the  cube  of  the  units. 


INVOLUTION  AND  EVOLUTION.  145 

206.  It  follows  from  Art.  205  that 

743  _  703  =  3  X.702  X  4  +  3  X  70  X  42  +  43 
=  (3  X  702  +  3  X  70  X  4  +  42)  X  4. 

That  is,  if  from  the  cube  of  any  number  of  two  figures  the 
cube  of  the  tens  be  subtracted,  the  remainder  is  equal  to  three 
times  the  square  of  the  tens,  plus  three  times  the  tens  times  the 
units,  plus  the  square  of  the  units,  multiplied  by  the  units. 

207.  VTe  have,  1^=1,  9^=729, 10^=1000,  99^=970299,  etc. 
That  is,  the  cube  of  any  number  of  one  figure  contains 

either  one,  two,  or  three  figures ;  the  cube  of  any  number  of 
two  figures  contains  either  four,  five,  or  six  figures  ;  etc. 

Hence,  if  a  point  be  placed  over  every  third  figure  of  any 
number,  beginning  at  the  units^  place,  the  number  of  points 
shows  the  number  of  figures  in  its  cube  root. 

208.  1.   Find  the  cube  root  of  74088. 


74088 

40  +  2 

403  = 

:  64000 

Trial  divisor,     3  x  40^  =  4800 

10088 

3x40x2=    240 

22=       4 

Complete  divisor,  5044 

10088 

Pointing  the  number  by  the  rule  of  Art.  207,  we  find  that  there  are 
two  figures  in  its  cube  root. 

The  greatest  perfect  cube  in  74  is  64,  which  is  the  cube  of  4  ;  then 
4  is  the  tens'  figure  of  the  root. 

Subtracting  the  cube  of  40,  or  64000,  from  74088,  the  remainder  is 
10088. 

By  Art.  206,  this  is  equal  to  3  x  402,  pi^g  3  x  40  x  the  units'  figure 
of  the  root,  plus  the  square  of  the  units'  figure,  multiplied  by  the  units' 
figure. 

•  That  is,  10088  is  equal  to  4800,  plus  3  x  40  x  the  units'  figure,  plus 
the  square  of  the  units'  figure,  multiplied  by  the  units'  figure. 

Then  to  obtain  the  units'  figure,  we  must  divide  10088  by  4800, 
plus  3  X  40  X  the  units'  figure,  plus  the  square  of  the  units'  figure. 


146  ARITHMETIC. 

Now  we  can  find  an  ajoproximate  value  of  the  units'  figure  by  divid- 
ing 10088  by  4800. 

The  quotient  is  2+  ;  we  then  infer  that  the  units'  figure  is  2. 

Adding  to  the  trial-divisor,  4800,  3  x  40  x  2,  or  240,  and  2^,  or  4, 
we  obtain  the  complete  divisor,  5044  ;  multiplying  this  by  2,  the  prod- 
act  is  10088,  which,  subtracted  from  10088,  leaves  no  remainder. 

Then  the  required  cube  root  is  42. 

Omitting  the  ciphers  for  the  sake  of  brevity,  and  condens- 
ing the  process,  it  will  stand  as  follows : 

74088  1^ 
64 


4800 

10088 

240 

4 
5044 

10088 

From  the  above  example,  we  derive  the  following  rule : 

Separate  the  number  into  periods  by  pohiting  every  third 
figure,  beginning  with  the  units^  place. 

Find  the  greatest  cube  in  the  left-hand  period^  and  write  its 
cube  root  as  the  first  figure  of  the  root;  subtract  the  cube  of  the 
first  root-figure  from,  the  first  period,  and  to  the  result  annex 
the  next  period. 

Divide  this  remainder  by  three  times  the  square  of  the  part 
of  the  root  already  found,  with  two  ciphers  annexed,  and  write 
the  quotient  as  the  next  figure  of  the  root. 

Add  to  the  trial-divisor  three  times  the  product  of  the  last 
root-figure  by  the  part  of  the  root  previously  found,  with  one 
cipher  annexed,  and  the  square  of  the  last  root-figure. 

Midtiply  the  complete  divisor  by  the  figure  of  the  root  last 
obtained,  and  subtract  the  product  from  the  remainder. 

If  other  periods  remain,  we  may  regard  the  part  of  the 
root  already  found  as  tens  with  respect  to  the  next  root- 
figure,  and  proceed  as  before ;  taking  three  times  the  square 
of  the  part  of  the  root  already  found,  with  two  ciphers 
annexed,  for  the  next  trial-divisor. 


INVOLUTION  AND  EVOLUTION.  147 

If  any  root-figure  is  0,  annex  two  ciphers  to  the  trial- 
divisor,  and  annex  to  the  remainder  the  next  period.  (Com- 
pare Ex.,  Art.  211.) 

If,  on  multiplying  any  complete  divisor  by  the  root-figure 
last  obtained,  the  product  is  greater  than  the  remainder,  the 
figure  of  the  root  last  obtained  is  too  great,  and  one  less 
must  be  substituted  for  it. 

2.   Find  the  cube  root  of  480048687. 


480048687 

783,  Ans. 

343 

14700 

137048 

1680 

64 

16444 

131552 

1825200 

5496687 

7020 

9 

183222 

9 

5496687 

209.   In  the  above  example,  the  first  complete  divisor  is 
3  X  70^  4-  3  X  70  X  8  4-  8^. 

The  second  trial-divisor  is  3  x  78^,  with  two  ciphers 
annexed. 

Now  by  Art.  197,  3x78^  =  3  x  (70^  +  2  x  70  x  8  -f  8^) 

=  3  X  702  +  6  X  70  X  8  +  3  X  82 
=  3  X  702  -I-  3  X  70  X  8  -f  82 
+  (3x70x8-f  2x82). 

That  is,  the  second  trial-divisor  may  be  ohtained  by  adding 
to  the  preceding  complete  divisor  the  second  number  and  twice 
the  third  number  required  to  form  it,  and  annexing  two  ciphers 
to  the  residt. 

The  above  rule  holds  for  any  trial-divisor,  and  will  be 
found  to  save  much  labor  in  extracting  cube  roots. 


148 


ARITHMETIC. 


210.  Cube  Root  of  Decimals. 

We  have  1.1«  =  1.331,  .25^  =  .015625,  etc. 

That  is,  the  cube  of  a  decimal  of  one  place  is  a  decimal  of 
three  places ;  the  cube  of  a  decimal  of  two  places  is  a  decimal 
of  six  places ;  etc. 

Hence,  if  a  point  he  placed  over  every  third  figure  of  any 
decimal  beginning  with  the  units'  place,  and  extending  in  either 
direction,  the  number  of  points  to  the  left  of  the  decimal  point 
shows  the  number  of  places  in  the  integral,  and  the  number  of 
points  to  the  right,  the  number  of  places  in  the  decimal,  portion 
of  the  cube  root. 

211.  The  rule  of  Art.  208  may  be  used  to  find  the  cube 
root  of  a  decimal,  provided  that  the  decimal  point  is  inserted 
in  its  proper  position  in  the  root. 

Example.     Find  the  cube  root  of  1073.741824. 


1073.741824 
1 


10.24,  Ans. 


30000 

600 

4 

30604 

73741 
61208 

600 

8 

3121200 

12240 

16 

3133456 

12533824 
12533824 

Since  300  is  not  contained  in  73,  the  second  root-figure  is  0. 

We  then  annex  two  ciphers  to  the  trial-divisor  300,  giving  30000, 
an«f  annex  to  the  remainder  the  next  period,  741. 

The  second  trial-divisor  is  obtained  by  the  rule  of  Art.  209  ;  adding 
to  the  preceding  complete  divisor,  30604,  the  first  number,  600,  and 
twice  the  second  number,  8,  required  to  form  it,  we  have  31212  ; 
annexing  two  ciphers  to  this,  the  result  is  3121200. 


INVOLUTION   AND  EVOLUTION. 


149 


EXAMPLES. 
212.   Find  the  cube  roots  of  the  following : 


1.  29791. 

2.  97.336. 

3.  .681472. 

4.  1860867. 
6.  1.481544. 
6.  .000941192. 


7.  8.242408. 

8.  51478848. 

9.  10077.696. 

10.  .517781627. 

11.  116.930169. 

12.  .031855013. 


13.  .724150792. 

14.  1039509.197. 

15.  .000152273304. 

16.  395446.904. 

17.  196.629718375. 

18.  .277550577667. 


213.  Cube  Root  of  an  Imperfect  Cube. 

If  there  is  a  final  remainder,  the  number  has  no  exact 
cube  root ;  but  we  may  obtain  an  approximate  value  of  the 
root,  correct  to  any  desired  number  of  decimal  places. 

1.   Find  the  cube  root  of  3  to  three  decimal  places. 


3.000000000 
1 


1.442+,  Ans. 


300 

120 

16 

436 

2000 
1744 

120 

32 
58800 

1680 

16 

60496 

256000 
241984 

1680 
32 

6220800 

14016000 

EXAMPLES. 
Find,  to  three  places  of  decimals,  the  cube  roots  of: 
2.   2.  3.    6.  4.    7.2.  5.   41.  6.    169. 


160  ARITHMETIC. 

To  find  the  approximate  cube  root  of  a  fraction  whose 
denominator  is,  and  whose  numerator  is  not,  a  perfect  cube, 
we  may  divide  the  approximate  cube  root  of  the  numerator 
by  the  cube  root  of  the  denominator. 

If  the  denominator  is  not  a  perfect  cube,  the  fraction 
should  be  reduced  to  an  equivalent  fraction  whose  denomi- 
nator is  a  perfect  cube. 

7.   Find  the  cube  root  of  i-  to  three  places  of  decimals. 
3/1      s/y      -VS     1.442+        .Q^^      . 


Find  the  cube  roots  of  the  following  to  three  places  of 
decimals : 

8.   f.  9.  f.  10.   ^.  11.   |.  12.   if. 

13.  Find  the  cube  root  of  ^-  to  four  places  of  decimals. 

14.  Find  the  cube  root  of  f|  to  four  places  of  decimals. 

214.  If  the  index  of  a  root  is  the  product  of  two  or  more 
numbers,  we  may  obtain  the  result  by  successive  extractions 
of  the  simpler  roots. 

Thus,  since  4  =  2x2,  we  may  find  the  fourth  root  of  a 
number  by  taking  the  square  root  of  its  square  root. 

Again,  since  6  =  3x2,  we  may  find  the  sixth  root  of  a 
number  by  taking  the  cube  root  of  its  square  root. 

EXAMPLES. 

1.  Find  the  fourth  root  of  1185921. 

2.  Find  the  fourth  root  of  33362176. 

3.  Find  the  fourth  root  of  59969536. 

4.  Find  the  fourth  root  of  69799526416. 

5.  Find  the  sixth  root  of  13841287201. 

6.  Find  the  sixth  root  of  75418890625. 


MENSURATION. 


151 


XV.    MENSURATION. 

215.   Mensuration  is  the  process  of  measuring  the  lengths 
of  lines,  the  areas  of  surfaces,  and  the  volumes  of  solids. 


PLANE  FIGURES. 

216.  The  Area  of  a  surface  is  the  number  of  times  that 
it  contains  another  surface,  taken  as  the  unit  of  measure- 
ment. 

Thus,  the  statement  that  the  area  of  a  surface  is  8  square 
inches,  signifies  that  a  square  inch  is  contained  in  the  sur- 
face 8  times. 

Two  surfaces  are  said  to  be  equivalent  when  they  have 
equal  areas. 

217.  An  Angle  is  the  figure  formed 
by  two  straight  lines  which  meet  at  a 
point ;  as  ABC 

The  point  of  meeting  B  is  called  the 
vertex  of  the  angle,  and  the  lines  AB  and 
BO  are  called  its  sides. 

A  right  angle  is  the  angle  formed  by 
two  straight  lines  which  are  perpendic- 
ular to  each  other :  as  DEF. 


218.  A  Polygon  is ,  a  plane  figure 
bounded  by  straight  lines ;  as  ABODE. 

The  bounding  lines,  AB,  BO,  etc.,  are 
called  the  sides  of  the  polygon,  and  their 
sum  is  called  the  perimeter. 

The   angles  of  the   polygon   are  the 
angles  ABO,  BOD,  etc.,  formed  by  the 
consecutive  sides ;  and  their  vertices  B,  0,  etc.,  are  called 
the  vertices  of  the  polygon. 

A  diagonal  is  a  line  joining  any  two  vertices  which  are 
not  consecutive  j  as  AG.  ^ 


152 


ARITHMETIC. 


219.  A  Regular  Polygon  is  one  whose 
sides  are  all  equal,  and  whose  angles  are 
all  equal ;  as  ABODE. 

The  point  0,  equally  distant  from  the 
vertices,  is  called  the  centre  of  the  poly- 
gon. 

220.  A  Triangle  is  a  polygon  of  three 
sides ;  as  ABC. 

The  base  of  the  triangle  is  the  side 
BC  on  which  it  is  supposed  to  stand; 
and  the  vertex  A,  opposite  to  the  base, 
is  called  the  vertex  of  the  triangle. 

Note.     Either  side  of  a  triangle  may  be  regarded  as  the  base. 

The  altitude  of  the  triangle  is  the  perpendicular  AD 
drawn  from  the  vertex  to  the  base. 

A  right  triangle  is  one  which  has  a 
right  angle ;  as  DEF,  which  has  a  right 
angle  at  F. 

The  side  DE,  opposite  to  the  right 
angle,  is  called  the  hypotenuse. 

221.  A  Quadrilateral  is  a  polygon  of 
four  sides. 

222.  A  Parallelogram  is  a  quadri- 
lateral whose  opposite  sides  are  parallel ; 
as  ABCD. 

The  base  of  the  parallelogram  is  the  side  AD  on  which  it 
is  supposed  to  stand ;  the  altitude  is  the  perpendicular  dis- 
tance EF  between  the  base  and  the  side  parallel  to  the  base. 


223.  A  Rectangle  is  a  parallelogram 
whose  angles  are  all  right  angles;  as 
ABCD. 

The  sides  AB  and  AD  are  called  the 
dimensions  of  the  rectangle. 


MENSURATION.  168 

A  Square  is  a  rectangle  whose  sides 
are  equal ;  as  EFGH. 

224.  A  Trapezoid  is  a  quadrilateral 
two  of  whose  sides  are  parallel;  as 
ABCD. 

The  parallel  sides  AD  and  BG  are 
called  the  bases  of  the  trapezoid ;  and 
the  perpendicular  distance  between 
them  EF  is  called  the  altitude.  J.  f  r> 

225.  Areas  of  Polygons. 

The  proofs  of  the  following  principles  may  be  found  in 
any  text-book  on  Geometry : 

1.  The  area  of  a  triangle  is  equal  to  one-half  the  product  of 
its  base  and  altitude. 

2.  The  area  of  a  parallelogram  (or  rectangle)  is  equal  to 
the  product  of  its  base  and  altitude. 

3.  The  area  of  a  square  is  equal  to  the  square  of  one  of  its 


F 

"^^~" 

Q 

E 

H 

B 

-E 

C 

\ 

1. 

\ 

4.  The  area  of  a  trapezoid  is  equal  to  one-half  the  sum  of 
its  bases,  multiplied  by  its  altitude. 

It  is  important  to  observe  that,  in  finding  the  product  of 
two  lines,  such  as  the  base  and  altitude  of  a  triangle,  their 
lengths  must  be  expressed  in  terms  of  the  same  unit,  and 
the  area  is  obtained  in  terms  of  the  square  of  this  unit. 

Thus,  to  multiply  3  feet  by  7  inches,  we  must  first  express 
3  feet  in  inches. 

Kow  3  feet  =  36  inches ;  and  multiplying  36  inches  by 
7  inches,  the  product  is  252  square  inches. 

EXAMPLES. 

226.  1.  Find  the  area  of  a  trapezoid  whose  bases  are 
43  in.  and  35  in.,  respectively,  and  altitude  21  in. 

By  Art.  225,  4,  the  area  of  the  trapezoid  is  J  x  (43  +  35)  x  21, 
or  819  sq.  in.,  Ans. 


164  ARITHMETIC. 

2.  Th^  area  of  a  triangle  is  1\  sq.  yd. ;  if  its  altitude  is 
40  in.,  what  is  its  base  in  feet  ? 

By  Art.  225,  1,  the  base  of  a  triangle  is  equal  to  twice  its  area, 
divided  by  its  altitude. 

We  have,  IJ  sq.  yd.  =  -\5  sq.  ft.,  and  40  in.  =  Y-  ft- 

Now  45^10^|^A  =  27^ 

2       3       2      ;p      4         ^ 

2 

Whence,  the  required  base  is  6|  ft.,  Ans. 

3.  What  is  the  side  of  a  square  whose  area  is  7^  sq.  rd.  ? 

By  Art.  225,  3,  the  side  of  a  square  is  equal  to  the  square  root  of 
its  area. 

Now  V7i=V^=|  =  2f. 

Hence,  the  required  side  is  2|  rd.,  Ans. 

4.  Find  the  area  in  square  inches  of  a  triangle  whose 
base  is  3^  it,  and  altitude  2^  ft. 

6.   Find  the  area  in  square  feet  and  square  inches  of  a 
square,  each  side  of  which  is  3  ft.  10  in. 

6.  Find  the  area  in  square  yards  of  a   parallelogram 
whose  base  is  7|-  ft.,  and  altitude  64  in. 

7.  Find  the  base  in  feet  of  a  triangle  whose  area  is  14|- 
sq.  ft.,  and  altitude  87  in. 

8.  Find  the  altitude  in  yards  of  a  rectangle  whose  area 
is  1782  sq.  ft.,  and  base  3  rd. 

9.  The  area  of  a  square  is  20  sq.  rd.  20  sq.  yd. ;  what  is 
its  side  in  feet  ? 

10.  Find  the  area  in  square  inches  of  a  trapezoid  whose 
bases  are  8|  ft.  and  5|-  ft.,  respectively,  and  altitude  3|  ft. 

11.  Find  the  altitude  in  yards  of  a  triangle  whose  area 
is  3  sq.  ft.,  and  base  27  in. 

12.  Find  the  base  in  inches  of  a  rectangle  whose  area  is 
3|-  sq.  yd.,  and  altitude  4|-  ft. 


MENSURATION.  155 

13.  Find  the  area  in  acres  of  a  square  field  whose  side  is 
396  ft. 

14.  Find  the  area  of  a  floor  whose  length  is  15  ft.  4  in., 
and  width  12  ft.  10  in. 

15.  A  triangular  house-lot  contains  3  acres.  If  its  base 
is  500  feet,  what  is  its  altitude  in  rods  ? 

16.  Find  the  width  of  a  rectangular  field  whose  area  is 
15  acres,  and  length  75  rods. 

17.  If  the  side  of  a  square  field  is  14  rd.  3  yd.,  how  much 
is  the  field  worth  at  $  242  an  acre  ? 

18.  Find  the  altitude  in  feet  of  a  trapezoid  whose  area  is 
139J  sq.  ft.,  and  bases  19  ft.  and  12  ft.,  respectively. 

19.  How  many  bricks,  each  8  inches  long  and  4^  inches 
wide,  will  be  required  to  lay  a  sidewalk  7^  feet  wide  and 
220  feet  long? 

20.  A  rectangular  garden,  64  feet  long  and  35  feet  wide, 
is  surrounded  by  a  walk  3  feet  wide.  How  many  square 
feet  are  there  in  the  walk  ? 

21.  A  map  is  1-J  ft.  long,  and  1  ft.  wide.  If  the  scale  of 
the  map  is  2^  miles  to  an  inch,  how  many  square  miles  of 
country  does  it  represent  ? 

22.  Find  the  length  in  yards  of  a  rectangular  field  whose 
area  is  7  acres,  and  width  363  feet. 

23.  Find  the  lower  base  in  rods  of  a  trapezoid  whose  area 
is  1  A.  50  sq.  rd.,  upper  base  99  yd.,  and  altitude  165  ft. 

24.  The  dimensions  of  a  rectangular  floor  are  22  ft.  6  in., 
and  16  ft.  9  in.  What  will  it  cost  to  cover  it  with  oil-cloth, 
at  $  1.52  a  square  yard  ? 

25.  How  many  paving-stones,  each  7  inches  long  and 
4  inches  wide,  will  be  required  to  pave  two  miles  of  street, 
63  ft.  in  width  ? 

26.  A  field  is  10  rd.  5  yd.  long,  and  13  rd.  3J  yd.  wide. 
How  much  is  it  worth  at  $  605  an  acre  ? 


156  ARITHMETIC. 

27.  A  man  sold  a  rectangular  field  for  3J  cents  a  square 
foot,  receiving  the  sum  of  $  5288.40 ;  if  the  field  was  28  rd. 
2  yd.  long,  what  was  its  width  ? 

28.  If  the  area  of  a  field  is  2  A.  32  sq.  rd.  26  sq.  yd.  2  sq. 
ft.,  and  its  width  15  rd.  2  yd.  2  ft.,  what  is  its  length  ? 

29.  A  field  is  18  rd.  3  yd.  2  ft.  wide,  and  33  rd.  1  yd.  1^ 
ft.  long.      How  much  is  it  worth  at  $  484  an  acre  ? 

227.  It  is  proved  in  Geometry  that 
In  a  right  triangle,  the  square  of  the  hypot- 
enuse is  equal  to  the  sum  of  the  squares  of  the 
other  two  sides. 

Note.     This  means  that,  if  the  sides  are  all  ex-  ^ 
pressed  in  terms  of  the  same  unit,  the  square  of  the 
length  of  the  hypotenuse  is  equal  to  the  sum  of  the  squares  of  the 
lengths  of  the  other  two  sides. 

It  follows  from  the  above  that 

In  a  right  triangle,  the  square  of  either  side  about  the  right 
angle  is  equal  to  the  square  of  the  hypotenuse,  minus  the  square 
of  the  other  side. 

EXAMPLES. 

228.  1.  The  sides  about  the  right  angle  of  a  right 
triangle  are  5  in.  and  1  ft.,  respectively;  find  the  hypotenuse 
in  inches. 

We  have  1  ft.  =  12  in. 


But  VP  +  12^  =  V25  +  144  =  V169  =  13. 

Whence,  the  hypotenuse  is  13  in.,  Ans. 

2.  The  hypotenuse  of  a  right  triangle  is  2f  in.,  and  one  of 
the  sides  about  the  right  angle  is  1\  in.  Find  the  other 
side. 


We  have  V(2|)2  -  (4)2  =  V(-V-)2  -  (|)2 

—  ■\/l3J'.  _  J_6   —   •v/225  _   15   —  2 1 

Whence,  the  required  side  is  2^  in.,  Ans. 


MENSURATION".  157 

3.  The  diagonal  of  a  square  is  2  feet ;  find  the  approxi- 
mate value  of  its  side  to  three  places  of  decimals. 

By  Art.  227,  the  square  of  the  diagonal  of  a  square  is  equal  to  twice 
the  square  of  its  side. 

Then  4  sq.  ft.  is  equal  to  twice  the  square  of  the  side. 
Hence,  tlie  side  is  equal  to  the  square  root  of  2  sq.  ft. 
The  approximate  value  of  V2  to  three  decimal  places  is  1.404. 
Whence,  the  required  side  is  1.404+  ft.,  Ajis. 

4.  The  sides  about  the  right  angle  of  a  right  triangle  are 
7  in.  and  2  ft.,  respectively.    Find  the  hypotenuse  in  inches. 

5.  The  sides  about  the  right  angle  of  a  right  triangle  ar^ 
1  rd.  31  yd.,  and  7  rd.  li  yd.,  respectively.  Find  the 
hypotenuse. 

6.  The  hypotenuse  of  a  right  triangle  is  3^  ft.,  and  one  of 
the  sides  about  the  right  angle  is  -|  yd.  Find  the  other  side 
in  inches. 

7.  The  hypotenuse  of  a  right  triangle  is  1  yd.  2  ft.  1  in., 
and  one  of  the  sides  about  the  right  angle  is  11  in.  Find 
the  other  side. 

8.  The  diagonal 'of  a  square  is  15  in.  Find  the  approxi- 
mate value  of  its  side  to  four  places  of  decimals. 

9.  The  sides  about  the  right  angle  of  a  right  triangle 
are  10  in.  and  7  in.,  respectively.  Find  the  approximate 
value  of  the  hypotenuse  to  four  places  of  decimals. 

10.  What  is  the  length  of  the  longest  straight  line  that 
can  be  drawn  on  a  floor  whose  length  is  17  ft.  8  in.,  and 
width  13  ft.  3  in.  ? 

11.  How  far  from  a  tower  35  ft.  high  must  the  foot  of  a 
ladder  37  ft.  long  be  placed,  so  as  to  exactly  reach  the  top 
of  the  tower  ? 

12.  A  tree  was  broken  off  12  ft.  above  the  ground,  and 
fell  so  that  its  top  lay  47  ft.  3  in.  from  the  foot  of  the  tree, 
the  end  where  it  was  broken  resting  on  the  stump.  What 
was  the  height  of  the  tree  ? 


158  ARITHMETIC, 

13.  A  vessel  sails  due  east  at  tlie  rate  of  6f  miles  an  hour, 
and  another  sails  due  south  at  the  rate  of  V2  miles  an  hour. 
How  far  apart  are  they  at  the  end  of  7  h.  45  min.  ? 

14.  A  ladder  37^  feet  long  is  placed  so  that  it  just 
reaches  a  window  22J  ft.  above  the  street ;  and  when  turned 
about  its  foot,  just  reaches  a  window  36  ft.  above  the  street 
on  the  other  side.     Find  the  width  of  the  street. 

15.  If  the  area  of  a  square  is  33  sq.  ft.  50  sq.  in.,  what  is 
the  length  of  its  diagonal  ? 

.  229.  A  Circle  is  a  portion  of  a  plane  bounded  by  a  curved 
line,  all  points  of  which  are  equally  distant  from  a  point 
within  called  the  centre;  as  ABD. 

The  bounding  curve  is  called  the  cir- 
cumference of  the  circle. 

A  radius  is  a  straight  line  drawn  from 
the  centre  to  the  circumference,  as  CD ;  a 
diameter  is  a  straight  line  drawn  through 
the  centre,  having  its  extremities  in  the 
circumference ;  as  AD. 

230.   Measurement  of  the  Circle. 

It  is  proved  in  Geometry  that,  approximately, 

1.  The  circumference  of  a  circle  is  equal  to  its  diameter 
multiplied  by  3.1416. 

2.  The  area  of  a  circle  is  equal  to  the  square  of  its  radius 
multiplied  by  3.1416. 

Note.  The  above  rules  give  the  circumference  in  terms  of  the  unit 
in  which  the  diameter  is  expressed,  and  tlie  area  in  terms  of  the  square 
of  the  unit  in  which  tlie  radius  is  expressed. 

The  following  rules  are  also  useful  ; 

3.  The  circumference  of  a  circle  is  approximately  equal  to 
twice  its  radius  multiplied  by  3.1416. 

4.  The  area  of  a  circle  is  approximately  equal  to  one-fourth 
the  square  of  its  diameter  multiplied  by  3.1416. 


MENSURATION.  159 

EXAMPLES. 

231.  1.  What  is  the  circumference  of  a  circle  whose 
diameter  is  7  inches  ? 

By  Art.  230,  1,  the  required  circumference  is 
7  X  3.1416  =  21.9912  in.,  Ans. 

2.  Eind  the  diameter  of  a  circle  whose  area  is  35  sq.  ft. 
By  Art.  230,  4,  the  square  of  the  diameter  of  a  circle  is  equal  to 

four  times  the  area  divided  by  3.1416. 

We  have  -^  =  44.5632+. 

o.l41o    ' 

The  square  root  of  44. 5632  +  is  6.67  + . 
Whence,  the  required  diameter  is  6.67+  ft.,  Ans. 

3.  The  radius  of  a  circle  is  7  inches.  Find  its  circumfer- 
ence and  area. 

4.  The  diameter  of  a  circle  is  50  ft.  Find  its  circumfer- 
ence in  yards. 

5.  The  circumference  of  a  circle  is  33  rods.  Find  its 
radius. 

6.  Find  the  diameter  in  inches  of  a  circle  whose  area  is 

one  square  foot. 

7.  If  the  diameter  of  the  earth  is  7912  miles,  what  is  the 
distance  around  it  ? 

8.  A  wheel  is  2  ft.  3  in.  in  diameter.  How  many  miles 
does  it  travel  in  revolving  2000  times  ? 

9.  How  many  acres  are  there  in  a  circular  field  whose 
diameter  is  500  feet  ? 

10.  A  horse  is  tied  by  a  rope  31  ft.  6  in.  long ;  over  how 
many  square  yards  of  ground  can  he  graze  ? 

11.  A  wheel  turns  29  times  in  travelling  154  yd.  2  ft. 
Find  its  diameter  in  inches. 

12.  The  floor  of  a  room  12  ft.  3  in.  long,  and  10  ft.  8  in. 
wide,  has  two  circular  openings  whose  radii  are  2  ft.  1  in.,  and 
1  ft.  5  in.,  respectively.     Find  the  area  of  floor  remaining. 


160  ARITHMETIC. 

13.  A  circular  pond,  100  ft.  in  diameter,  is  surrounded 
by  a  walk  4  feet  wide.     Find  the  area  of  the  walk. 

14.  If  a  wheel  is  5  feet  in  diameter,  how  many  times  does 
it  revolve  in  running  27  miles  ? 

15.  The  diameter  of  a  circle  is  10  inches.  What  is  the 
side  of  an  equivalent  square  (Art.  216)  ? 

16.  The  side  of  a  square  is  8  feet.  What  is  the  circum- 
ference of  an  equivalent  circle  ? 

17.  Two  plots  of  ground,  one  a  square,  the  other  a  circle, 
contain  each  70,686  sq.  ft.  How  much  longer  is  the  perim- 
eter of  the  square  than  the  circumference  of  the  circle  ? 

SOLIDS. 

232.  The  volume  of  a  solid  is  the  number  of  times  that 
it  contains  another  solid,  adopted  as  the  unit  of  measure- 
ment. 

Thus,  the  statement  that  the  volume  of  a  solid  is  6  cu.  ft., 
means  that  a  cubic  foot  is  contained  in  the  solid  6  times. 

Two  solids  are  said  to  be  equivalent  when  they  have  equal 
volumes. 

233.  A  Polyedron  is  a  solid  bounded  by  planes. 

The  bounding  planes  are  called  the /aces  of  the  polyedron; 
their  intersections  are  called  the  edges,  and  the  intersections 
of  the  edges  are  called  the  vertices. 

234.  A  Prism  is  a  polyedron  two  of  whose  faces  are  equal 
and  parallel,  the  other  faces  being  parallelograms ;  as  A-I. 

The  equal  and  parallel  faces,  ABCDE 
and  FQHIK,  are  called  the  bases  of  the 
prism,  and  the  remaining  faces  the 
lateral  faces. 

The  lateral  faces  taken  together  form 
the  lateral  surface  of  the  prism ;  and 
their  intersections,  AF,  BG,  etc.,  are 
called  the  lateral  edges. 


MENSURATION. 


161 


/]— 

/ 

dI 

— -}C 


The  lateral  area  is  the  area  of  the  lateral  surface. 
The  altitude  is  the  perpendicular  distance  LM  between 
the  planes  of  the  bases. 

235.  A  Right  Prism  is  one  whose 
lateral  edges  are  perpendicular  to  its 
bases;  as  ABG-DEF. 

The  lateral  faces  are  rectangles,  and  the 
lateral  edges  are  equal  to  the  altitude. 


236.  A  Rectangular  Parallelepiped  is 

a  right  prism  whose  six  faces  are  all 
rectangles ;  as  A-G. 

The  dimensions  are  the  three  edges 
which  meet  at  any  vertex. 

A  Cube  is  a  rectangular  parallelopiped 
whose  six  faces  are  all  squares. 


237.  A  Pyramid  is  a  polyedron  bounded  by  a  polygon 
and  a  series  of  triangles  having  a  com- 
mon vertex ;  as  0-ABCD. 

The  polygon  ABGD  is  called  the  base 
of  the  pyramid ;  and  the  common  vertex 
0  of  the  triangular  faces  is  called  the 
vertex. 

The  triangular  faces  are  called  the 
lateral  faces,  and  taken  together  form  the  lateral  surface. 

The  intersections  OA,  OB,  etc.,  of  the  lateral  faces  are 
called  the  lateral  edges;  and  the  area  of  the  lateral  surface 
is  called  the  lateral  area. 

The  altitude  is  the  perpendicular  OE 
drawn  from  the  vertex  to  the  base. 

238.  A  Regular  Pyramid  is  one  whose 
base  is  a  regular  polygon,  and  whose 
vertex  lies  in  the  perpendicular  erected 
at  the  centre  of  the  base  j  as  0-ABCD, 


162 


ARITHMETIC. 


The  slant  height  of  a  regular  pyramid  is  the  altitude  of 
any  one  of  its  lateral  faces ;  that  is,  it  is  the  straight  line 
drawn  from  the  vertex  to  the  middle  point  of  any  side  of 
the  base :  as  OF. 


239.  A  Frustum  of  a  pyramid  is  that 
portion  of  a  pyramid  included  between 
the  base  and  a  plane  parallel  to  the  base ; 
as  ABC-DEF. 

The  altitude  of  the  frustum  is  the  per- 
pendicular distance  between  the  planes 
of  its  bases  ;  as  GH. 

240.  The  slant  height  of  a  frustum  of 
a  regular  pyramid  is  the  straight  line 
joining  the  middle  points  of  the  parallel 
sides  of  any  lateral  face ;  as  LM. 


■D'K Z^ 

/  ^ /i 

V  Je^-^^ 

^ 

°I\!A' 

/  1 

M 

1 



~U 

241.  Lateral 
Polyedrons. 


Areas  and  Volumes  of 


The  proofs  of  the  following  principles  may  be  found  in 
any  text-book  on  Solid  Geometry : 

1.  The  lateral  area  of  a  right  prism  (or  rectangular  paral- 
lelopiped)  is  equal  to  the  perimeter  of  its  base  multiplied  by  its 
altitude. 

2.  The  volume  of  a  prism  (or  rectangular  parallelopiped) 
is  equal  to  the  area  of  its  base  multiplied  by  its  altitude. 

3.  The  volume  of  a  cube  is  equal  to  the  cube  of  one  of  its 
edges. 

4.  The  lateral  area  of  a  regular  pyramid  is  equal  to  the 
perimeter  of  its  base  multiplied  by  one-half  its  slant  height. 

5.  The  volume  of  a  pyramid  is  equal  to  the  area  of  its  base 
midtiplied  by  one-third  its  altitude. 

6.  The  lateral  area  of  a  frustum  of  a  regular  pyramid  is 
equal  to  one-half  the  sum  of  the  perimeters  of  its  bases,  multi- 
plied by  its  slant  height. 


MENSURATION.  163 

7.  The  volume  of  a  frustum  of  a  pyramid  is  equal  to  the 
sum  of  the  areas  of  its  bases,  plus  the  square  root  of  the  product 
of  the  areas  of  its  bases,  multiplied  by  one-third  its  altitude. 

To  multiply  an  area  by  a  length,  the  area  must  be  expressed 
in  terms  of  the  square  of  the  unit  in  which  the  length  is 
expressed,  and  the  product  is  obtained  in  terms  of  the  cube 
of  this  unit. 

Thus,  to  multiply  2  sq.  ft.  by  10  in.,  we  must  first  express 
the  2  sq.  ft.  in  square  inches. 

Now  2  sq.  ft.  =  288  sq.  in. ;  and  multiplying  288  sq.  in. 
by  10  in.,  the  product  is  2880  cu.  in. 

EXAMPLES. 

242.  1.  Find  the  lateral  area  of  a  regular  pyramid,  the 
perimeter  of  whose  base  is  17  in.,  and  slant  height  8  in. 

By  Art.  241,  4,  the  required  lateral  area  is 

J  X  17  X  8,  or  68  sq.  in.,  Ans. 

2.  The  volume  of  a  rectangular  parallelepiped  is  693  cu. 
ft.,  and  the  dimensions  of  its  base  are  11  ft.  and  7  ft.  Find 
its  altitude,  and  the  area  of  its  entire  surface. 

The  area  of  the  base  is  7  x  11,  or  77  sq.  ft. 

By  Art.  241,  2,  the  altitude  of  the  parallelepiped  is  equal  to  its 
volume,  divided  by  the  area  of  its  base. 

Hence,  the  required  altitude  =  -^^^/-,  or  9  ft. 
By  Art.  241,  1,  the  lateral  area  is  equal  to 

2  X  (7  +  11)  X  9,  or  324  sq.  ft. 

The  area  of  the  two  bases  is  2  x  77,  or  154  sq.  ft. 
Hence,  the  area  of  the  entire  surface  is 

324  sq.  ft.  +  154  sq.  ft.,  or  478  sq.  ft.,  Ans. 

3.  Find  the  volume  of  a  frustum  of  a  pyramid  whose 
lower  base  is  a  square  7  in.  on  a  side,  upper  base  4  in.  on  a 
side,  and  altitude  6  in. 

The  areas  of  the  bases  are  49  sq.  in.  and  16  sq.  in.,  respectively. 
Now,  V49  x  16  =  V78i  =  28. 


164  ARITHxMETIC. 

Hence,  the  square  root  of  the  product  of  the  areas  of  the  bases  is 
28  sq.  in. 

Then  by  Art.  241,  7,  the  required  volume  is 

^  X  (49  +  16  +  28)  X  6,  or  186  cu.  in.,  Ans. 

4.  Find  the  lateral  area  and  volume  of  a  prism  whose 
altitude  is  11  in.,  having  for  its  base  a  right  triangle  whose 
sides  are  5  in.,  12  in.,  and  13  in. 

5.  Find  the  volume  and  area  of  the  entire  surface  of  a 
cube  whose  edge  is  3:^  inches. 

6.  Find  the  lateral  area  of  a  regular  pyramid  whose  base 
is  a  square  6  ft.  on  a  side,  and  slant  height  12  feet. 

7.  The  volume  of  a  prism,  whose  base  is  a  square,  is 
637  cu.  ft.,  and  its  altitude  is  13  ft.  Find  the  length  of 
each  side  of  the  base,  and  the  lateral  area. 

8.  What  is  the  volume  of  a  pyramid  whose  altitude  is  21 
in.,  having  for  its  base  a  right  triangle  whose  sides  are  8  in., 
15  in.,  &,nd  17  in.  ? 

9.  Find  the  lateral  area  of  a  frustum  of  a  regular  pyra- 
mid whose  lower  base  is  a  square  9  ft.  on  a  side,  upper  base 
5  ft.  on  a  side,  and  slant  height  14  ft. 

10.  The  volume  of  a  pyramid,  whose  base  is  a  square,  is 
847  cu.  in.,  and  its  altitude  is  21  in.  Find  the  length  of 
each  side  of  the  base. 

11.  Find  the  volume  of  a  frustum  of  a  pyramid  whose 
lower  base  is  a  rectangle  15  in.  by  6  in.,  upper  base  5  in.  by 
2  in.,  and  altitude  15  in. 

12.  The  lateral  area  of  a  regular  pyramid  is  1680  sq.  in. 
The  base  is  a  triangle  whose  sides  are  all  equal,  and  the 
slant  height  is  35  in.  Find  the  length  of  each  side  of  the 
base. 

13.  A  box  is  13  in.  long,  12  in.  wide,  and  7  in.  deep. 
Find  its  volume,  and  the  area  of  its  entire  surface. 

14.  The  volume  of  a  box  is  84  cu.  ft.,  and  the  dimensions 
of  its  bottom  are  7  ft.  and  4  ft.  Find  its  depth,  and  the  area 
of  its  entire  surface. 


MENSURATION.  165 

15.  A  wagon  7  ft.  long  and  4  ft.  wide  is  piled  with  wood 
to  a  depth  of  5^  ft.  What  is  the  value  of  the  wood  at 
$7.04  a  cord? 

16.  The  lateral  area  of  a  frustum  of  a  regular  pyramid  is 
936  sq.  in.  The  lower  base  is  a  square  18  in.  on  a  side,  and 
the  upper  base  is  6  in.  on  a  side.  Find  the  slant  height  of 
the  frustum. 

17.  The  volume  of  a  cube  is  4^  cu.  ft.  What  is  the  area 
of  its  entire  surface  in  square  inches  ? 

18.  What  will  be  the  cost  of  a  pile  of  wood  35|-  ft.  long, 
6i  ft.  high,  and  4  ft.  wide,  at  $  7.68  a  cord  ? 

19.  A  monument  whose  height  is  12  ft.,  is  in  the  form  of 
a  pyramid  with  a  square  base,  2  ft.  lOi  in.  on  a  side.  Find 
its  weight,  at  180  lb.  to  the  cubic  foot. 

20.  How  many  bricks,  each  8  in.  long,  2J  in.  wide,  and 

2  in.  thick,  will  be  required  to  build  a  wall  18  ft.  long,  3  ft. 
high,  and  11  in.  thick  ? 

21.  What  must  be  the  length  of  a  pile  of  wood  that  is 

3  ft.  9  in.  wide,  and  5  ft.  4  in.  high,  to  contain  5  cords  ? 

22.  A  monument  is  in  the  form  of  a  frustum  of  a  square 
pyramid  8  ft.  in  height,  surmounted  by  a  square  pyramid 
2  ft.  in  height.  If  each  side  of  the  lower  base  of  the  frus- 
tum is  3  ft.,  and  each  side  of  the  upper  base  2  ft.,  find  the 
volume  of  the  monument. 

23.  The  volume  of  a  frustum  of  a  pyramid  is  210  cu.  in. 
The  lower  base  is  a  right  triangle  whose  sides  are  6  in.,  8  in., 
and  10  in.,  and  the  upper  base  a  right  triangle  whose  sides 
are  3  in.,  4  in.,  and  5  in.     Find  the  altitude  of  the  frustum. 

24.  A  box  made  of  2  in.  plank,  without  a  cover,  measures 
on  the  outside  3  ft.  2  in.  long,  2  ft.  3  in.  wide,  and  1  ft.  6 
in.  deep.  How  many  cubic  feet  of  material  were  used  in 
its  construction  ? 

25.  The  base  of  a  square  pyramid  is  14  in.  on  a  side,  and 
the  altitude  is  24  in.     Find  its  lateral  area  and  volume. 


166 


ARITHMETIC. 


26.  The  water  in  a  certain  ditch  flows  at  the  rate  of  2j 
miles  an  hour.  If  the  ditch  is  3  ft.  wide,  and  2  ft.  deep, 
how  many  cubic  feet  of  water  pass  through  it  in  one  day  ? 

243.  A  Cylinder  is  a  solid  bounded  by  two  parallel  circles, 
and  a  curved  surface  all  points  of  which 
are  equally  distant  from  a  straight  line 
within  called  the  axis;  as  A-D. 

The  parallel  circles  AB  and  CD  are 
called  the  bases  of  the  cylinder,  and  the 
curved  surface  is  called  the  lateral 
surface. 

The  lateral  area  is  the  area  of  the  lateral  surfacCj  and  the 
altitude  is  the  perpendicular  distance  EF  between  the  bases. 

244.  A  Cone  is  a  solid  bounded  by  a  circle,  and  a  curved 
surface  which  tapers  uniformly  to  a  point  called  the  vertex; 
as  OAB. 

The  circle  AB  is  called  the  base  of 
the  cone,  and  the  curved  surface  is 
called  the  lateral  surface. 

The  lateral  area  is  the  area  of  the 
lateral  surface,  and  the  altitude  is  the 
perpendicular  OC  from  the  vertex  to 
the  base. 

The  slant  height  is  the  straight  line  drawn  from  the  vertex 
to  any  point  in  the  circumference  of  the  base ;  as  OD. 

245.  A  Frustum  of  a  cone  is  that  portion  of  a  cone  in- 
cluded between  the  base  and  a  plane  parallel  to  the  base ; 
as  A-D. 

The  altitude  of  the  frustum  is  the 
perpendicular  distance  between  the 
bases ;  as  EF. 

The  slant  height  is  that  portion  of 
the  slant  height  of  the  cone  included 
between  the  bases  of  the  frustum ;  as 
GH. 


MENSURATION.  167 

246.  A  Sphere  is  a  solid  bounded  by  a  curved  surface, 
all  points  of  which  are  equally  distant 

from  a  point  within  called  the  centre  ; 
as  ABD. 

A  radius  is  a  straight  line  drawn 
from  the  centre  to  the  surface,  as  OA  ; 
a  diameter  is  a  straight  line  drawn 
through  the  centre,  having  its  extremi- 
ties in  the  surface,  as  AC. 

247.  Measurement  of  the  Cylinder,  Cone,  and  Sphere. 
It  is  proved  in  Geometry  that : 

1.  The  lateral  area  of  a  cylinder  is  equal  to  the  circumfer- 
ence of  its  base  multiplied  by  its  altitude ;  or,  approximately, 
to  twice  the  radius  of  its  base,  times  its  altitude,  times  3.1416. 

21  The  volume  of  a  cylinder  is  equal  to  the  area  of  its  base 
multiplied  by  its  altitude;  or,  approximately,  to  the  square  of 
the  radius  of  its  base,  times  its  altitude,  times  3.1416. 

3.  The  lateral  area  of  a  cone  is  equal  to  the  circumference 
of  its  base  multiplied  by  one-half  its  slant  height;  or,  approxi- 
mately, to  the  radius  of  its  base,  times  its  slant  height,  times 
3.1416. 

4.  The  volume  of  a  cone  is  equal  to  the  area  of  its  base 
multiplied  by  one-third  its  altitude;  or,  approximately,  to  the 
square  of  the  radius  of  its  base,  times  one-third  its  altitude, 
times  3.1416. 

5.  TTie  lateral  area  of  a  frustum  of  a  cone  is  equal  to  one- 
half  the  sum  of  the  circumferences  of  its  bases,  multiplied  by 
its  slant  height;  or,  approximately,  to  the  sum  of  the  radii  of 
its  bases,  times  its  slant  height,  times  3.1416. 

6.  TTie  volume  of  a  frustum  of  a  cone  is  equal  to  the  sum 
of  the  areas  of  its  bases,  plus  the  square  root  of  the  product  of 
the  areas  of  its  bases,  multiplied  by  one-third  its  altitude; 
or,  approximately,  to  the  sum  of  the  squares  of  the  radii  of 
its  bases,  plus  the  product  of  the  radii  of  its  bases,  times  one- 
third  its  altitude,  times  3.1416. 


168  ARITHMETIC. 

Also,  approximately, 

7.  The  area  of  a  sphere  is  equal  to  the  square  of  its  di- 
ameter, or  four  times  the  square  of  its  radius,  multiplied  by 
3.1416. 

8.  The  volume  of  a  sphere  is  equal  to  one-sixth  the  cube  of  its 
diameter,  or  four-thirds  the  cube  of  its  radius,  multiplied  by 
3.1416. 

Note.  The  second  paragraph  on  page  163  applies  with  equal  force 
to  the  above  rules. 

EXAMPLES. 

248.  1.  Find  the  lateral  area  and  volume  of  a  cylinder 
whose  altitude  is  9  in.,  and  radius  of  base  4  in. 

By  Art.  247,  1,  the  required  lateral  area  is 

8  X  9  X  3.1416,  or  226.1952  sq.  in. 
By  Art.  247,  2,  the  required  volume  is 

16  X  9  X  3.1416,  or  452.3904  cu.  in. 

2.  Find  the  radius  of  a  sphere  whose  area  is  452.3904 
sq.  in. 

By  Art.  247,  7,  the  square  of  the  radius  of  a  sphere  is  equal  to  its 
area  divided  by  4  times  3.1416,  or  12.5664. 
Hence,  the  square  of  the  required  radius  is 
452:3904  ^^3,        ^^^ 
12.5664 
The  square  root  of  36  is  6  ;  whence  the  required  radius  is  6  in.,  Ans. 

3.  The  volume  of  a  cone  is  1005.312  cu.  ft.,  and  the 
radius  of  its  base  is  8  ft.  Find  its  altitude  and  slant 
height. 

By  Art.  247,  4,  the  altitude  of  a  cone  is  equal  to  three  times  its 
volume,  divided  by  3.1416  times  the  square  of  the  radius  of  its  base. 

Hence,  the  required  altitude  is ,  or  15  ft. 

3.1416x64 

In  the  right  triangle  ABD,  by  Art.  227,  the  square 
of  AB is  equal  to  the  sum  of  the  squares  of  AD  and  BD. 

But  82  +  152  =  64  +  225  =  289,  which  is  the  square 
of  17. 

Hence,  the  required  slant  height  is  17  ft.  u 


MENSURATION.  .  169 

4.  What  is  the  volume  of  a  frustum  of  a  cone  whose 
altitude  is  12  in.,  and  radii  of  bases  6  in.  and  2  in.,  respec- 
tively? 

By  Art.  247,  6,  the  required  volume  is 

(36  +  4+12)  X  4  X  3.1416,  or  663.4528  cu.  in.,  Ans. 

6.  Find  the  lateral  area  and  volume  of  a  cylinder  whose 
altitude  is  7  in.,  and  radius  of  base  3  in. 

*^   6.   Find  the  area  and  volume  of  a  sphere  whose  radius 
is  4  in. 

7.  Find  the  lateral  area  of  a  frustum  of  a  cone  whose 
slant  height  is  14  ft.,  and  radii  of  bases  9  ft.  and  2  ft., 
respectively. 

8.  Find  the  lateral  "area  and  volume  of  a  cone  whose 
altitude  is  12  in.,  and  radius  of  base  5  in. 

9.  Find  the  volume  of  a  frustum  of  a  cone  whose  alti- 
tude is  9  ft.,  and  radii  of  bases  10  ft.  and  6  ft.,  respectively. 

10.  The  lateral  area  of  a  cone  is  188.496  sq.  in.,  and  the 
radius  of  its  base  is  6  in.    Find  its  slant  height  and  altitude. 

11.  The  area  of  a  sphere  is  314.16  sq.  in.  Find  its  di- 
ameter and  volume. 

12.  The  volume  of  a  cylinder  is  2412.7488  cu.  in.,  and 
its  altitude  is  12  in.     Find  the  radius  of  its  base. 

13.  The  lateral  area  of  a  frustum  of  a  cone  is  603.1872 
sq.  in.,  and  the  radii  of  its  bases  are  5  in.  and  11  in.,  respec- 
tively.    Find  its  slant  height. 

14.  Assuming  the  earth  to  be  a  sphere  7900  miles  in 
diameter,  find  its  area  and  volume. 

15.  A  tent,  in  the  shape  of  a  cone,  has  a  slant  height  of 
16  feet,  and  a  diameter  at  the  base  of  24  feet.  How  many 
square  yards  of  material  were  used  in  its  construction  ? 

16.  The  volume  of  a  frustum  of  a  cone  is  779.1168  cu.  ft., 
and  the  radii  of  its  bases  are  10  ft.  and  2  ft.,  respectively. 
Find  its  altitude. 


170  •  ARITHMETIC. 

17.  What  will  it  cost  to  gild  a  ball  25  inches  in  diameter, 
at  $  13.50  a  square  foot  ? 

18.  The  altitude  of  a  cone  is  9  in.,  and  the  radius  of  its 
base  is  7  in.  Find  the  altitude  of  an  equivalent  cylinder 
(Art.  232),  the  diameter  of  whose  base  is  10  in. 

19.  How  many  cubic  feet  are  there  in  a  log  of  wood  20 
feet  long,  whose  girth  is  3  feet  ? 

20.  A  basin  is  in  the  shape  of  a  hemisphere  whose 
diameter  is  2f  yards.  How  many  cubic  feet  of  water  will 
it  hold? 

21.  How  many  cubic  feet  of  metal  are  there  in  a  hollow 
iron  tube  18  ft.  long,  whose  outer  diameter  is  7  in.,  and 
thickness  1  in.  ? 

22.  Find  the  radius  of  a  sphere,  whose  surface  is  equiv- 
alent to  the  lateral  surface  of  a  cylinder,  whose  altitude  is 
8  ft.,  and  radius  of  base  4  ft. 

23.  The  volume  of  a  sphere  is  7238.2464  cu.  in.  Find  its 
radius. 

24.  A  cylindrical  vessel,  8  in.  in  diameter,  is  filled  to  the 
brim  with  water.  A  ball  is  immersed  in  it,  displacing  water 
to  the  depth  of  2^  in.     Find  the  diameter  of  the  ball. 

25.  The  outer  diameter  of  a  spherical  shell  is  9  in.,  and 
its  thickness  is  1  in.  What  is  its  weight,  if  a  cubic  inch  of 
the  metal  weighs  i  lb.  ? 

26.  How  many  cubic  feet  are  there  in  a  column  whose 
length  is  22  ft.,  diameter  of  larger  end  10  in.,  and  diameter 
of  smaller  end  7  in.  ? 

27.  The  altitude  of  a  frustum  of  a  cone  is  6  ft.,  and  the 
radii  of  its  bases  are  3  ft.  and  2  ft.,  respectively.  What 
is  the  diameter  of  an  equivalent  sphere  ? 

28.  If  a  gallon  contains  231  cu.  in.,  what  must  be  the 
depth  of  a  cylindrical  measure,  3  in.  in  diameter,  to  hold  a 
quart  ? 


MENSURATION.  171 

APPLICATIONS    OF    MENSURATION. 

249.  Capacity  of  Bins,  Tanks,  and  Cisterns. 

The  following  equivalents  are  to  be  used  in  the  examples 
of  the  present  article  : 

1  bushel  =  2150.42  cu.  in.  1    gallon  =  231  cu.  in. 

1  bushel  =  1\  cu.  ft.  7|  gallons  =  1  cu.  ft. 

1.  How  many  bushels  of  grain  can  be  put  into  a  bin  5  ft. 
3  in.  long,  3  ft.  6  in.  wide,  and  4  ft.  1  in.  deep  ? 

We  have 
5  ft.  3  in.  =  63 in.,  3  ft.  6  in.  =  42  in.,  and  4  ft.  1  in.  =  49  in. 

Then  the  volume  of  the  bin  =  63  x  42  x  49,  or  129654  cu.  in. 

Since  one  bushel  contains  2150.42  cu.  in.,  as  many  bushels  can  be 
put  into  the  bin  as  2150.42  is  contained  times  in  129654,  which  is 
60.29+,  Ans. 

EXAMPLES. 

2.  How  many  bushels  of  grain  can  be  put  into  a  bin  7  ft. 
long,  3  ft.  wide,  and  4  ft.  deep  ? 

3.  If  a  ton  of  coal  occupies  38  cu.  ft.,  how  many  tons  can 
be  put  into  a  bin  8  ft.  long,  5^  ft.  wide,  and  6^  ft.  deep  ? 

4.  How  many  gallons  will  a  tank  hold  which  is  2  ft.  6  in. 
long,  2  ft.  wide,  and  1  ft.  9  in.  deep  ? 

5.  How  many  gallons  of  water  can  be  put  into  a  cylindrical 
tank  whose  diameter  is  50  in.,  and  depth  65  in.  ? 

6.  How  deep  must  a  bin  be  that  is  6  ft.  long  and  4  ft. 
wide,  to  hold  84  bushels  of  grain  ? 

7.  What  must  be  the  depth  of  a  cubical  bin  to  hold  100 
bushels  of  wheat  ? 

•  8.   If  a  tank  is  66  inches  wide  and  42  inches  deep,  how 
long  must  it  be  to  hold  1000  gallons  ? 

9.  How  deep  must  a  cistern  be,  whose  diameter  is  60  in., 
to  hold  800  gallons  ? 


172  ARITHMETIC. 

10.  How  many  bushels  of  oats  can  be  put  into  a  bin  4  ft. 
7  in.  long,  3  ft.  5  in.  wide,  and  3  ft.  10  in.  deep  ? 

11.  If  a  tank  8  ft.  long,  5  ft.  wide,  and  3  ft.  deep,  is  tilled 
with  oil,  how  much  is  the  oil  worth  at  13  cents  a  gallon  ? 

12.  What  must  be  the  diameter  of  a  cylindrical  tank, 
whose  depth  is  55  inches,  to  hold  640  gallons  ? 

13.  How  deep  must  a  bin  be  that  is  7  ft.  2  in.  long,  and 
5  ft.  7  in.  wide,  to  hold  150  bushels  of  rye  ? 

14.  A  cubical  bin,  5  ft.  3  in.  deep,  is  filled  with  wheat. 
What  is  its  value  at  $0.96  per  bushel  ? 

15.  To  what  depth  must  a  cistern  38  inches  in  diameter 
be  filled,  to  hold  304  gallons  ? 

16.  A  well  is  3  ft.  in  diameter,  and  32  ft.  deep.  How 
many  barrels  of  31|-  gallons  each  will  it  contain  ? 

17.  How  many  bushels  can  be  put  into  a  cylindrical  re- 
ceptacle whose  diameter  is  3  ft.  6  in.,  and  depth  5  ft.  4  in.  ? 

18.  What  must  be  the  diameter  of  a  cistern,  of  depth  4 
feet,  to  hold  400  gallons  ? 

19.  If  a  tank  holds  500  gallons  of  water,  how  many 
bushels  of  grain  can  be  put  into  it  ? 

250.  Carpeting  Rooms. 

1.  A  floor  is  15  ft.  4  in.  long,  and  11  ft.  9  in.  wide. 
How  much  will  it  cost  to  cover  it  with  carpeting,  each 
length  2  ft.  8  in.  wide,  at  63  cents  a  yard,  no  allowance 
being  made  for  waste  in  matching  the  pattern  ? 

If  the  strips  are  laid  lengthioise  of  the  room,  as  many  strips  will  be 
required  as  2  ft.  8  in.  is  contained  times  in  11  ft.  9  in. 

32  in.  is  contained  in  141  in.  4  times,  with  a  remainder  of  13  in.     • 

Then  Jive  strips  will  be  required. 

The  total  length  of  the  five  strips  is  15  ft.  4  in.  x  5,  or  76  ft.  8  in.  ; 
that  is,  15 1  yd. 

Then  the  cost  will  be  15|  x  |0.63,  or  $9.80,  Ans. 


MENSURATION.  173 

Note.  It  will  be  understood,  in  the  following  examples,  that  the 
strips  are  laid  lengthwise  of  the  room,  unless  the  contrary  is  specified. 

EXAMPLES. 

2.  How  many  yards  of  carpeting  |-  of  a  yard  wide  will  be 
required  for  a  floor  17  ft.  long  and  14  ft.  wide  ? 

3.  How  much  will  it  cost  to  cover  a  floor  15  ft.  long  and 
11  ft.  3  in.  wide  with  oil-cloth,  at  40  cents  a  square  yard  ? 

4.  How  many  yards  of  carpeting  2  ft.  7  in.  wide  will  be 
required  for  a  floor  20  ft.  11  in.  long  and  16  ft.  4  in.  wide, 
if  the  strips  run  lengthwise  of  the  room  ?  How  many  if  the 
strips  run  across  the  room  ? 

5.  How  much  will  it  cost  to  cover  a  floor  14  ft.  3  in. 
square  with  straw  matting,  in  strips  one  yard  wide,  at  44 
cents  a  yard  ? 

6.  A  floor  is  16  ft.  11  in.  long  and  12  ft.  1  in.  wide. 
How  much  will  it  cost  to  cover  it  with  carpeting,  each 
length  2  ft.  5  in.  wide,  at  75  cents  a  yard  ? 

7.  How  many  yards  of  carpeting  2  ft.  10  in.  wide,  will 
be  required  for  a  floor  16  ft.  9  in.  long  and  13  ft.  8  in.  wide, 
if  there  is  a  waste  of  4|-  inches  in  each  strip  in  matching  the 
pattern  ? 

8.  Which  way  should  the  strips  run  to  carpet  most  eco- 
nomically a  floor  18  ft.  3  in.  long  and  15  ft.  4  in.  wide,  the 
strips  being  2  ft.  9  in.  wide  ? 

9.  How  much  will  it  cost  to  cover  a  floor  21  ft.  6  in.  long 
and  18  ft.  4  in.  wide  with  carpeting  2  ft.  6  in.  wide,  at  87 
cents  a  yard,  if  there  is  a  waste  of  |-  of  a  yard  in  each  strip 
in  matching  the  pattern  ? 

10.  Which  will  be  the  cheaper,  to  cover  a  floor  19  ft.  7  in. 
long  and  14  ft.  9  in.  wide  with  matting,  in  strips  2  ft.  8  in. 
wide,  laid  lengthwise,  at  42  cents  a  yard,  or  to  cover  it  with 
oil-cloth  at  48  cents  a  square  yard  ? 


174  ARITHMETIC. 

251.  Plastering  and  Papering. 

1.  How  much  will  it  cost  to  plaster  a  room  16  ft.  long, 
13  ft.  wide,  and  9  ft.  high,  at  40  cents  a  square  yard,  allow- 
ing 64  sq.  ft.  for  doors  and  windows  ? 

The  area  of  the  four  walls  is  2  x  (16  +  13)  x  9,  or  522  sq.  ft. 
The  area  of  the  ceihng  is  16  x  13,  or  208  sq.  ft. 
Then  the  total  area  to  be  plastered  is  522  +  208  -  64,  or  666  sq.  ft.  ; 
that  is,  74  sq.  yd. 

Hence,  at  40  cents  a  square  yard,  the  total  cost  will  be 
74  X  $0.40=  $29.60,  ^ns. 

2.  How  many  rolls  of  paper,  IJ  ft.  wide,  9  yards  to  a 
roll,  will  be  required  to  paper  a  room  19  ft.  long,  14  ft.  wide, 
and  9J  ft.  high,  allowing  for  two  doors,  each  3  ft.  wide  and 
7J  ft.  high,  four  windows,  each  3  ft.  wide  and  5^  ft.  high, 
and  a  base-board  9  in.  wide  ? 

The  area  of  the  four  walls  is  2  x  (19  +  14)  x  9^^,  or  605  sq.  ft. 

The  area  of  the  two  doors  is  2  x  3  x  7J,  or  44  sq.  ft. 

The  area  of  the  four  windows  is  4  x  3  x  5 J,  or  66  sq.  ft. 

The  length  of  the  base-board  is  the  distance  around  the  room,  less 
the  width  of  the  two  doors  ;  that  is,  6Q  —  6,  or  60  ft. 

Then  the  area  of  the  base-board  is  60  x  f ,  or  45  sq.  ft. 

Thus,  the  total  area  to  be  deducted  from  the  area  of  the  four  walls 
is  44  +  66  +  45,  or  155  sq.  ft. 

Hence,  the  area  to  be  papered  is  605  —  155,  or  450  sq.  ft. 

The  area  of  each  roll  is  If  x  27,  or  45  sq.  ft. 

Then,  as  many  rolls  will  be  required  as  46  is  contained  times  in 
450  ;  that  is,  10  rolls,  Ans. 

EXAMPLES. 

3.  How  much  will  it  cost  to  plaster  a  room  18  ft.  long, 
15  ft.  wide,  and  10  ft.  high,  at  39  cents  a  square  yard,  allow- 
ing 102  sq.  ft.  for  doors  and  windows  ? 

4.  How  much  will  it  cost  to  plaster  a  room  20  ft.  8  in. 
long,  16  ft.  3  in.  wide,  and  9  ft.  6  in.  high,  at  48  cents  a 
square  yard,  allowing  128  sq.  ft.  36  sq.  in.  for  doors  and 
windows  ? 


MENSURATION.  175 

5.  How  many  rolls  of  paper,  1|  ft.  wide,  12  yards  to  a 
roll,  will  be  required  to  paper  a  room  17  ft.  long,  12  ft. 
wide,  and  9  ft.  high,  no  allowance  being  made  for  doors  or 
windows  ? 

6.  How  much  will  it  cost  to  paper  a  room  14  ft.  square, 
and  8|-  ft.  high,  with  paper  1  ft.  10  in.  wide,  10  yards  to  a 
roll,  at  75  cents  a  roll;  allowing  80  sq.  ft.  for  doors  and 
windows  ? 

7.  A  room  16  ft.  long,  12  ft.  wide,  and  9  ft.  high,  has 
three  doors,  each  3  ft.  wide  and  7  ft.  high,  two  windows, 
each  3  ft.  wide  and  5  ft.  9  in.  high,  and  a  base-board  9 
inches  wide.  How  much  will  it  cost  to  plaster  it  at  44 
cents  a  square  yard  ? 

8.  How  much  will  it  cost  to  paper  a  room  15  ft.  long, 
11  ft.  wide,  and  10  ft.  high,  with  paper  1  ft.  11  in.  wide,  11 
yards  to  a  roll,  at  $  1.10  a  roll,  allowing  175  sq.  ft.  for  doors, 
windows,  and  base-board  ? 

9.  What  will  it  cost  to  plaster  a  hemispherical  dome, 
whose  diameter  is  60  ft.,  at  50  cents  a  square  yard  ? 

10.  How  many  rolls  of  paper,  1  ft.  11^  in.  wide,  12  yards 
to  a  roll,  will  be  required  to  paper  a  room  21  ft.  long,  15^ 
ft.  wide,  and  8f  ft.  high,  allowing  for  three  doors,  each  3  ft. 
wide  and  6f  ft.  high,  three  windows,  each  2f  ft.  wide  and 
5i  ft.  high,  and  a  base-board  1  ft.  wide  ? 

11.  A  room  23  ft.  6  in.  long,  15  ft.  4  in.  wide,  and  9  ft. 
9  in.  high,  has  three  doors,  each  3  ft.  wide  and  6  ft.  8  in. 
high,  four  windows,  each  2  ft.  9  in.  wide  and  5  ft.  9  in.  high, 
and  is  surrounded  by  a  base-board  9  in.  wide.  How  much 
will  it  cost  to  plaster  it  at  36  cents  a  square  yard  ? 

12.  Find  the  cost  of  papering  a  room  18  ft.  4  in.  long,  14 
ft.  6  in.  wide,  •  and  9  ft.  6  in.  high,  with  paper  1  ft.  9  in. 
wide,  111  yards  to  a  roll,  at  $  1.19  a  roll;  allowing  for  two 
doors,  each  3  ft.  wide  and  7  ft.  high,  and  two  windows, 
each  2  ft.  9  in.  wide'  and  6  ft.  high  ? 


176  ARITHMETIC. 

252.  Board  Measure. 

A  board  one  inch  or  less  in  thickness  is  said  fco  have  as 
many  Board  Feet  as  there  are  square  feet  in  its  surface. 

If  it  is  more  than  an  inch  in  thickness,  the  number  of 
board  feet  is  found  by  multiplying  the  number  of  square 
feet  in  its  surface  by  the  number-  of  inches  in  its  thickness. 

In  measuring  a  board  that  tapers,  the  width  is  taken  as 
one-half  the  sum  of  the  widths  of  the  two  ends. 

Boards  are  usually  sold  at  a  certain  price  per  hundred 
(C.)  or  per  thousand  (M.)  board  feet. 

1.  Find  the  cost  of  24  planks,  each  22  ft.  8  in.  long,  21 
in.  wide,  and  2  J  in.  thick,  at  $  25  per  M. 

22  ft.  8  in.  =  %8-  ft.,  21  in.  =  |  ft.,  and  2^  in.  =  f  in. 
Then  the  total  number  of  board  feet  is 
17  3       6 

^  X  ^  X  ^  X  ?^,  or  2142. 

3      ^     ^ 

At  $25  per  M.,  the  total  cost  will  be 

2.142  X  $25,  or  $53.55,  Ans. 

EXAMPLES. 

2.  Find  the  number  of  board  feet  in  a  board  16  ft.  6  in. 
long,  14  in.  wide,  and  1  in.  thick. 

3.  Find  the  number  of  board  feet  in  a  board  10  ft.  long, 
11  in.  wide,  and  f  in.  thick. 

4.  Find  the  number  of  board  feet  in  a  piece  of  timber 
25  ft.  9  in.  long,  9  in.  wide,  and  8  in.  thick. 

5.  Find  the  number  of  board  feet  in  a  plank  18  ft.  8  in. 
long,  1  ft.  5  in.  wide,  and  3^  in.  thick. 

6.  Find  the  number  of  board  feet  in  a  tapering  plank  15 
ft.  4  in.  long,  2  ft.  3  in.  wide  at  one  end,  and  1  ft.  11  in.  wide 
at  the  other,  and  3f  in.  thick. 

7.  Find  the  cost  of  45  spruce  joists,  each  14  ft.  long, 
6  in.  wide,  and  4  in.  thick,  at  $  14  per  M. 


MENSURATION.  177 

8.  Find  the  cost  of  150  boards,  each  11  ft.  8  in.  long, 
5  in.  wide,  and  |-  in.  thick,  at  $  18.30  per  M. 

9.  Find  the  cost  of  30  planks,  each  17  ft.  4  in.  long,  1  ft. 
10  in.  wide,  and  2f  in.  thick,  at  $  2.55  per  C. 

10.  Find  the  cost  of  a  plank-walk  75  ft.  long,  2  ft.  6  in. 
wide,  and  |  in.  thick,  at  $  31.50  per  M. 

11.  Find  the  cost  of  75  planks,  each  12  ft.  10  in.  long, 
1  ft.  7  in.  wide  at  one  end,  and  1  ft.  1  in.  wide  at  the  other, 
and  If  in.  thick,  at  $  15  per  M. 

253.   Measurement  of  Hound  Timber. 

To  find  the  side  of  the  squared  timber  that  can  be  sawed 
from  a  log,  multiply  the  diameter  of  the  smaller  end  by  .707. 

To  find  the  number  of  board  feet  in  the  squared  timber 
that  can  be  sawed  from  a  log,  multiply  together  one-half  the 
length  in  feet,  the  diameter  of  the  smaller  end  in  feet,  arid  the 
diameter  of  the  smaller  end  in  inches. 

1.  Find  the  side,  and  the  number  of  board  feet,  in  the 
squared  timber  that  can  be  sawed  from  a  log  whose  length 
is  15^  ft.,  and  diameter  at  the  smaller  end  18  in. 

By  the  first  of  the  above  rules,  the  side  is 

18  in.  X  .707,  or  12.726  in. 
By  the  second  rule,  the  number  of  board  feet  is 

i  X  15i  X  -  X  18  =  1  X  —  X  -  X  X^  =§15  =  204|. 

2  '2  2^2  4  ^ 

EXAMPLES. 

Find  the  side,  and  the  number  of  board  feet,  in  the  squared 
timber  that  can  be  sawed  from  a  log  whose  length  is : 

2.  18  ft.,  and  diameter  1  ft. 

3.  21  ft.,  and  diameter  of  smaller  end  1\  ft. 

4.  17^  ft.,  and  diameter  15  in. 

5.  15  ft.  9  in.,  and  diameter  1  ft.  2  in. 

6.  23  ft.  10  in.,  and  diameter  of  smaller  end  1  ft.  7  in. 


178  ARITHMETIC. 

254.   Specific  Gravity. 

The  Specific  Gravity  of  a  substance  is  the  number  of  times 
that  the  weight  of  a  certain  portion  of  the  substance  con- 
tains the  weight  of  an  equal  bulk  of  water. 

For  example,  a  cubic  foot  of  copper  weighs  8.8  times  as 
much  as  a  cubic  foot  of  water ;  hence,  the  specific  gravity 
of  copper  is  8.8. 

In  the  following  examples,  the  weight  of  a  cubic  foot  of 
water  is  taken  as  1000  oz.,  or  62.5  lb. 

1.  What  is  the  weight  of  a  cubic  foot  of  iron,  if  its 
specific  gravity  is  7.53  ? 

Since  a  cubic  foot  of  water  weighs  62.5  lb.,  and  iron  is  7.53  times 
as  heavy  as  water,  a  cubic  foot  of  iron  will  weigh 

62.5  lb.  X  7.53,  or  470.625  lb.,  Ans. 

2.  A  mass  of  granite  (specific  gravity  2.6)  weighs  7800 
lb. ;  how  many  cubic  feet  does  it  contain  ? 

Since  one  cubic  foot  of  granite  weighs  62.5  lb.  x  2.6,  or  162.5  lb., 
to  weigh  7800  lb,  will  take  as  many  cubic  feet  of  granite  as  162.5  is 
contained  times  is  7800. 

Dividing  7800  by  162.5,  the  result  is  48  cu.  ft.,  Ans. 


EXAMPLES. 

3.  Find  the  weight  in  pounds  of  a  cubic  foot  of  copper 
(specific  gravity  8.81) . 

4.  Find  the  w^eight  in  pounds  of  a  cubic  yard  of  brick- 
work (specific  gravity  1.8). 

5.  Find  the  weight  in  pounds  of  5  cu.  ft.  288  cu.  in.  of 
yellow  pine  (specific  gravity  .46). 

6.  Find  the  weight  in  ounces  of   a  cubic  inch  of   mer- 
cury (specific  gravity  13.596). 

7.  If  a  mass  of  iron  (specific  gravity  7.68)  "weighs  6  T., 
how  many  cubic  feet  does  it  contain  ? 


MENSURATION. 


179 


8.  If  a  mass  of  tin  (specific  gravity  7.5)  weighs  18750 
lb.,  how  many  cubic  feet  does  it  contain  ? 

9.  If  a  certain  bulk  of  alcohol  (specific  gravity  .791) 
weighs  197f  oz.,  how  many  cubic  inches  does  it  contain  ? 

10.  If  a  piece  of  gold  (specific  gravity  19.4)  weighs  37 
lb.  14  oz.  4  dr.,  how  many  cubic  inches  does  it  contain  ? 

11.  If  a  cubic  foot  of  glass  weighs  170  lb.,  find  its  spe- 
cific gravity. 

12.  If  a  cubic  yard  of  oak  weighs  1485  lb.,  find  its  specific 
gravity. 

13.  If  3  cu.  ft.  432  cu.  in.  of  silver  weighs  2132  lb.  13  oz. 
avoirdupois,  find  its  specific  gravity. 

14.  If  a  cubic  inch  of  brass  weighs  77.2864  dr.,  find  its 
specific  gravity. 

255.  Geometrical  Explanation  of  Square  and  Cube  Root. 
Square  Root. 

Let  it  be  required  to  find  the  square  root  of  1296. 

Let  A  CEG  be  a  square  containing  1296  sq.  in. 

To  find  its  side  in  inches. 

Since  a  square  whose  side  is 
30  in.  contains  900  sq.  in.,  and 
a  square  whose  side  is  40  in. 
contains  1600  sq.  in.,  the  side 
of  the  given  square  must  be 
between  30  and  40  in. 

Thus  the  tens'  figure  of  the 
root  is  3.  . 


Fig.  1 
D 


Fig.  2 


K 

P 

B 

30 
30 

Q 

H 


Fig.  3 


p 

Q 

R 

30 


30 


M 


N 


Removing  from  the  given 
square  the  square  ABKH, 
whose  side  is  30  in.,  there 
remains  an  irregular  figure, 
shown  in  Fig.  2,  composed  of  two  rectangles  P  and  Q,  and  a  square 
i?,  whose  united  area  is  1296  —  900,  or  396  sq.  in. 

The  rectangles  and  the  square  may  be  arranged  as  shown  in  Fig.  3, 
forming  a  rectangle  LM,  whose  altitude  is  the  units'  figure  of  the  root. 

Now  the  altitude  of  a  rectangle  is  equal  to  its  area  divided  by  its  base. 


180 


ARITHMETIC. 


Since  the  base  of  each  of  the  rectangles  P  and  Q  is  30  in.,  the  base 
of  the  rectangle  LM  is  something  more  than  60  in. 

If  we  divide  the  area  of  LM,  396  sq.  in.,  by  its  approximate  base, 
60  in.,  we  obtain  something  more  than  6  in.  as  the  approximate  alti- 
tude. 

If,  now,  we  make  trial  of  6  in.  as  the  altitude  of  the  rectangle,  the 
base  ZriVis  60  in.  +  6  in.,  or  66  in.  ;  and  multiplying  this  by  the  alti- 
tude, 6  in.,  the  result  is  396  sq.  in. 

But  this  is  just  the  area  of  the  irregular  figure  of  Fig.  2. 

We  then  conclude  that  the  units'  figure  of  the  root  is  6 ;  whence, 
the  required  root  is  30  +  6,  or  36. 

The  above  process  is  exactly  in  accordance  with  the  Rule  of  Art.  200. 


Cube  Eoot. 

Let  it  be  required  to  find  the  cube  root  of  13824. 


Fig.  i 


Fig.  2 


A 

/ 

) — 

/ 

/ 

/    / 

7 

/ 

20 

/ 
E 

F 

/ 
/ 

/ 

«/ 

«.    / 

Let  ABhe  s,  cube  containing  13824  cu.  in. 

To  find  its  edge  in  inches. 

Since  a  cube  whose  edge  is  20  in.  contains  8000  cu.  in,,  and  a  cube 
whose  edge  is  30  in.  contains  27000  cu.  in.,  the  edge  of  the  given  cube 
must  be  between  20  and  30  in. 

Thus  the  tens'  figure  of  the  root  is  2. 

Removing  from  the  given  cube  a  cube  whose  edge  is  20  in.,  there 
remains  an  irregular  solid  CD,  whose  volume  is  13824  -8000,  or  5824 
cu.  in. 

Removing  from  CD  the  three  solids  E,  F,  and  6r,  there  remains  an 
irregular  solid,  shown  in  Fig.  3,  composed  of  three  rectangular  paral- 
lelepipeds, H,  K,  and  i,  and  a  cube. 

The  solids  E,  F,  6r,  H,  K,  and  i,  and  the  cube  M,  may  be  arranged 
as  shown  in  Fig.  4,  forming  an  irregular  solid  iVP,  whose  altitude  is 
the  units'  figure  of  the  root. 

Now  the  altitude  of  this  solid  is  equal  to  its  volume  divided  by  the 
area  of  its  base. 


MENSURATION.  181 

Since  the  area  of  the  base  of  each  of  the  solids  E,  F,  and  G  is  20^, 
or  400  sq.  in.,  the  sum  of  the  areas  of  their  bases  is  3  x  400,  or  1200 
sq.  in. 


Fig. 

% 

./ 

/ 

/    //// 

E 

F 

G 

H 

K 

L 

M 

N    210  20  20        4      444 

Then  the  area  of  the  base  of  the  solid  NP  is  something  more  than 
1200  sq.  in. 

If  we  divide  the  volume  of  iVP,  5824  cu.  in.,  by  its  approximate 
area  of  base,  1200  sq.  in.,  we  obtain  something  more  than  4  in.  as  the 
approximate  altitude. 

If,  now,  we  make  trial  of  4  in.  as  the  altitude  of  the  solid,  the  area 
of  the  base  of  each  of  the  solids  H,  K,  and  i  is  4  x  20,  or  80  sq.  in., 
and  the  sum  of  the  areas  of  their  bases  is  3  x  80,  or  240  sq.  in. 

Also,  the  area  of  the  base  of  the  cube  M  is  4^,  or  i6  sq.  in. 

Then  the  area  of  the  base  of  the  solid  NP  is  1200  +  240  +  16,  or 
1456  sq.  in. ;  and  multiplying  this  by  the  altitude,  4  in.,  the  result  is 
6824  cu.  in. 

But  this  is  just  the  volume  of  the  irregular  solid  of  Fig.  2. 

We  then  conclude  that  the  units'  figure  of  the  root  is  4  ;  whence, 
the  required  root  is  20  +  4,  or  24. 

The  above  process  is  exactly  in  accordance  with  the  rule  of  Art.  208. 

PROBLEMS  IN  MENSURATION  INVOLVING  THE  METRIC 
SYSTEM. 

Note.  The  remainder  of  the  present  chapter  may  be  omitted  by 
those  who  have  not  previously  taken  the  chapter  on  the  Metric  System. 

256.  Mensuration  of  Plane  Figures. 

1.  Find  the  area  in  square  decimeters  of  a  rectangle 
whose  base  is  8.9™  and  altitude  735'^'". 

2.  Find  the  altitude  in  meters  of  a  triangle  whose  area 
is  16.8«'J^%  and  base  .096^'". 

3.  The  hypotenuse  of  a  right  triangle  is  41™,  and  one  of 
the  sides  about  the  right  angle  is  .9^".  Find  the  other  side 
in  hektometers. 


182  ARITHMETIC. 

4.  Find  the  base  in  dekameters  of  a  parallelogram  whose 
area  is  89.1'"»*=™,  and  altitude  SS*"™. 

5.  Find  the  circumference  in  meters  of  a  circle  whose 
radius  is  23^'". 

6.  Find  the  area  in  square  hektometers  of  a  triangle 
whose  base  is  528'*™  and  altitude  S-SQ"""™. 

7.  The  sides  about  the  right  angle  of  a  right  triangle 
are  IS*"  and  SS**™,  respectively.  Find  the  hypotenuse  in 
dekameters. 

8.  The  diameter  of  a  circle  is  IS**".  Find  its  area  in 
square  centimeters. 

9.  Find  the  side  in  decimeters  of  a  square  whose  area  is 
143641**1™'". 

10.  Find  the  area  in  square  decimeters  of  a  trapezoid 
whose  bases  are  3.51™  and  4852™™,  respectively,  and  altitude 
.0295^™. 

11.  Find  the  radius  in  dekameters  of  a  circle  whose  area 
is  .19635«'i«™. 

12.  Find  the  diameter  in  centimeters  of  a  circle  whose 
circumference  is  12™™. 

13.  Find  the  lower  base  in  centimeters  of  a  trapezoid 
whose  area  is  3.6"^™,  upper  base  .13°™,  and  altitude  18^™. 

14.  Find  the  area  in  ars  of  a  rectangular  field  253.8'*™  long 
and  13.9™  wide. 

15.  Find  the  area  in  centars  of  a  floor  .5498*^™  long  and 
467'=™  wide. 

16.  A  circular  grass-plot,  17™  in  diameter,  is  surrounded 
by  a  walk  18'*™  wide.     Find  the  area  of  the  walk  in  centars. 

17.  A  triangular  house-lot  contains  .241853^*.  If  its  base 
is  7.34*^™,  what  is  its  altitude  in  meters  ? 

18.  The  area  of  a  square  field  is  77.2641"^  Find  its  side 
in  dekameters. 


MENSURATION.  183 

19.  What  is  the  length  of  the  longest  straight  line  that 
can  be  drawn  in  a  rectangular  field  whose  length  is  204™  and 
width  85™? 

20.  If  the  diameter  of  a  wheel  is  75*^™,  how  many  times 
will  it  revolve  in  travelling  12.9591^'"  ? 

21.  The  side  of  a  sq  uire  field  is  87.2™  How  much  is  the 
field  worth  at  $  8750  a  hektar  ? 

22.  A  vessel  sails  due  north  at  the  rate  of  9.3^™  an  hour, 
and  another  sails  due  west  at  the  rate  of  12.4^"'  an  hour. 
How  far  apart  are  they  at  the  end  of  4  h.  12  min.  ? 

23.  Two  circles  whose  radii  are  25^™  and  173'"  have  the 
samfe  centre.  How  many  square  hektometers  of  area  are 
included  between  their  circumferences  ? 

24.  How  many  acres  are  there  in  a  rectangular  field  whose 
length  is  8.9^™,  and  width  .75^"  ? 

25.  A  cow,  tied  by  a  rope  to  a  stake,  can  graze  over 
415.4766^''  ™  of  ground.     What  is  the  length  of  the  rope  ? 

26.  Find  the  diagonal  in  hektometers  of  a  square  whose 
area  is  .1764'^^". 

27.  A  man  sold  a  rectangular  field  for  28  cents  a  centar, 
receiving  the  sum  of  $  915.04.  If  the  field  was  .43°™  wide, 
what  was  its  length  in  dekameters  ? 

28.  The  floor  of  a  room  is  6™  long  and  43^™  wide,  and  has 
a  circular  opening  .17''"  in  diameter.  Find  the  number  of 
square  centimeters  in  the  floor. 

29.  A  rectangular  garden  is  surrounded  by  a  walk  1.2™ 
wide,  containing  248.16^*^™.  If  the  garden  is  64'"  long,  what 
is  its  width  ? 

30.  A  circular  field  contains  a  hektar.  What  is  its  di- 
ameter in  dekameters  ? 

31.  The  diagonal  of  a  square  is  76™.  Find  the  approxi- 
mate length  of  its  side. 


184  ARITHMETIC. 

Mensuration  of  Solids. 

1.  Find  the  lateral  area  and  volume  of  a  prism  whose 
altitude  is  7'^'",  having  for  its  base  a  right  triangle  whose 
sides  are  3^"",  4^"\  and  5^"". 

2.  Find  the  area  and  volume  of  a  sphere  whose  radius  is 

3cm^ 

3.  Find  the  lateral  area  and  volume  of  a  cone  whose 
altitude  is  24™,  and  radius  of  base  7™. 

4.  The  volume  of  a  rectangular  parallelopiped  is  1920"="  ^"\ 
and  the  dimensions  of  its  base  are  15*^™  and  8^"\  Find  its 
altitude  and  the  area  of  its  entire  surface. 

5.  The  volume  of  a  pyramid  is  1320'="^".  The  base  is  a 
right  triangle  whose  sides  are  10^"^,  24^™,  and  26^™.  Find 
the  altitude  of  the  pyramid. 

6.  Find  the  lateral  area  of  a  frustum  of  a  cone,  whose 
slant  height  is  8™,  radius  of  lower  base  9™,  and  radius  of 
upper  base  3™. 

7.  Theareaof  a  sphere  is  201.0624^^^™.  Find  its  radius 
and  volume. 

8.  The  base  of  a  regular  pyramid  is  a  square  whose 
area  is  67.24''^*^'",  and  its  slant  height  is  579'™.  Find  its 
lateral  area  in  square  meters. 

9.  Find  the  volume  of  a  frustum  of  a  pyramid  whose 
lower  base  is  a  rectangle  12^™  by  3°™,  upper  base  8^™  by 
2°'",  and  altitude  16.8^™. 

10.  The  volume  of  a  frustum  of  a  cone  is  1894.3848'="™ 
and  the  radii  of  its  bases  are  11™  and  5™.    Find  its  altitude. 

11.  Find  the  lateral  area  in  square  millimeters,  and  the 
volume  in  cubic  centimeters,  of  a  cylinder  whose  altitude  is 
81™™,  and  radius  of  base  9^=™. 

12.  The  lateral  area  of  a  cone  is  565.488"^  ™,  and  the  radius 
of  its  base  is  12™.  Find  its  slant  height,  altitude,  and 
volume,  "^ 


MENSURATION.  185 

13.  Find  the  lateral  area  of  a  frustum  of  a  regular  pyra- 
mid whose  lower  base  is, a  square  15™  on  a  side,  upper  base 
9™  on  a  side,  and  slant  height  8.49™. 

14.  A  basin  is  in  the  shape  of  a  hemisphere  whose 
diameter  is  4.3*^™.  How  many  deciliters  of  water  will  it 
contain  ?     What  is  the  weight  of  this  water  in  decigrams  ? 

15.  A  wood  pile  is  2.5'"  long,  13*^™  wide,  and  125*=™  high. 
How  much  is  it  worth  at  $  3.36  a  ster  ? 

16.  What  will  be  the  cost  of  gilding  a  ball  65*^™  in  diam- 
eter, at  $1.50  a  square  decimeter  ? 

17.  A  room  is  5™  long,  48^^™  wide,  and  326^™  high.  How 
many  dekaliters  of  air  does  it  contain  ? 

18.  How  much  will  it  cost  to  dig  a  ditch  7°™  long,  18*^™ 
wide,  and  12**™  deep,  at  75  cents  a  ster  ? 

19.  A  wood  pile,  19"^™  long  and  142^=™  wide,  contains 
3.72324^*.     Find  its  height  in  meters. 

20.  Find  the  number  of  cubic  meters  in  a  tapering  piece 
of  timber  9™  long,  one  of  which  is  5*^™  square,  and  the  other 
32*=™  square. 

21.  A  bar  of  iron  is  6.2™  long,  8^™  wide,  and  13™™  thick. 
Find  its  weight  in  kilograms,  if  iron  is  7.53  times  as  heavy 
as  water. 

22.  A  cylindrical  boiler  is  4™  long,  and  13'^°'  in  diameter. 
How  many  liters  of  water  will  it  contain  ? 

23.  A  trench  is  42™  long,  8*^™  deep,  2.1™  wide  at  the 
top,  and  1.7™  wide  at  the  bottom.  How  many  hektoliters 
of  water  will  it  contain  ? 

24.  How  many  bricks,  each  2^™  long,  82™™  wide,  and  4.8*=™ 
thick,  will  be  required  to  build  a  wall  37™  long,  41*'™  wide, 
and  19.2*^™  high  ? 

25.  A  spherical  shell,  5*=™  thick,  has  an  outside  diameter 
of  4*™.  How  many  cubic  centimeters  of  metal  does  it  con- 
tain? 


186  ARITHMETIC. 

26.  A  brass  rod  is  3.5™  long  and  4^""  in  diameter.  Find 
its  weight  in  dekagrams,  if  brass  is  8.3  times  as  heavy  as 
water. 

27.  Find  the  weight  in  kilograms  of  the  water  that  can 
be  put  into  a  pail  in  the  shape  of  a  frustum  of  a  cone,  whose 
depth  is  24*=™,  diameter  at  the  top  32""',  and  diameter  at  the 
bottom  22*=™. 

28.  A  cannon-ball,  23^""  in  diameter,  is  dropped  into  a 
cubical  box  filled  with  water,  whose  depth  is  23*="".  How 
many  centiliters  of  water  will  be  left  in  the  box  ? 

29.  A  hopper,  in  the  form  of  an  inverted  frustum  of  a 
pyramid,  holds  1.364^^^  of  grain.  It  is  1™  square  at  the  top, 
and  2*^™  square  at  the  bottom.     Find  its  depth  in  meters. 

30.  A  projectile  consists  of  two  hemispheres,  connected 
by  a  cylinder.  If  the  altitude  and  diameter  of  the  cylinder 
are  2*^™  and  13^°',  respectively,  find  the  number  of  cubic 
decimeters  in  the  projectile. 

Capacity  of  Bins,  Tanks,  and  Cisterns,  Carpeting,  Plas- 
tering, and  Papering. 

1.  A  tank  is  3.7""  long,  .98^"  wide,  and  1.6™  deep.  How 
many  liters  of  water  will  it  contain  ? 

2.  How  many  hektoliters  of  wheat  can  be  put  into  a 
bin  2.3™  long,  12^™  wide,  and  18*^™  deep  ? 

3.  How  many  meters  of  carpeting  8^™  wide  will  be  re- 
quired for  a  floor  6.59™  long  and  5™  wide  ? 

4.  How  much  will  it  cost  to  plaster  a  room  5™  long,  4™ 
wide,  and  2.9™  high,  at  45  cents  a  square  meter,  allowing 
j^Q  gsqm  jpQj.  (joors  and  windows  ? 

5.  What  must  be  the  depth  in  meters  of  a  tank  24^™ 
long  and  78''™  wide,  to  hold  35.9424™  of  water  ? 

6.  What  must  be  the  length  in  decimeters  of  a  bin  1.88" 
deep  and  .13^™  wide,  to  hold  657.436°^  of  grain  ? 


MENSURATION.  187 

7.  A  cylindrical  cistern  whose  diameter  is  3""  is  filled 
with,  water  to  a  depth  of  19*^"".  How  many  hektoliters  of 
water  does  it  contain,  and  what  is  the  weight  of  the  water 
in  hektograms  ? 

8.  How  many  rolls  of  paper  7^"^  wide,  ll*"  to  a  roll,  will 
be  required  to  paper  a  room  5.8'°  long,  4.7™  wide,  and  2.8" 
high,  allowing  8.75'^™  for  doors  and  windows  ? 

9.  A  cubical  tank  holds  491.3^^  of  water.  What  is  its 
depth  in  meters  ? 

10.  A  hemispherical  dome  is  25^  in  diameter.  How  much 
will  it  cost  to  plaster  it  at  48  cents  a  square  meter  ? 

11.  What  will  it  cost  to  cover  a  floor  7.9™  long  and  5.8™ 
wide  with  carpeting  78*""  wide,  at  95  cents  a  meter,  if  there 
is  a  waste  of  5*^™  in  each  strip  in  matching  the  pattern  ? 

12.  A  tank  5.1™  long,  2.3"^  wide,  and  1.7™  deep,  is  filled 
by  a  pipe  through  which  pass  289*^^  of  water  a  minute.  How 
long  will  it  take  to  fill  it  ? 

13.  How  many  rolls  of  paper  68*=™  wide,  10™  to  a  roll,  will 
be  required  to  paper  a  room  7™  long,  5.5™  wide,  and  3.2™ 
high,  with  two  doors,  each  88*=™  wide  and  22*^™  high,  and  four 
windows,  each  86*=™  wide  and  17.5*^™  high  ? 

14.  A  well  is  13*^™  in  diameter,  and  12™  deep.  How  many 
kiloliters  of  water  will  it  hold  ? 

15.  A  cylindrical  tank,  16*^™  deep,  holds  15393.84^2  of 
water.     What  is  its  diameter  in  meters  ? 

16.  How  much  will  it  cost  to  plaster  a  room  7.2™  long, 
4.9™  wide,  and  3™  high,  at  42  cents  a  square  meter,  allow- 
ance being  made  for  two  doors,  each  9^™  wide  and  21*^™  high, 
three  windows,  each  8*^™  wide  and  18^™  high,  and  a  base- 
board 2*^™  wide  ? 

17.  If  a  metric  ton  of  coal  occupies  1.184*="™,  how  many 
metric  tons  can  be  put  into  a  bin  37^™  long,  12**™  wide,  and 
36^™  deep  ? 


188  ARITHMETIC. 

18.  Which  way  should  the  strips  run  to  carpet  most 
economically  a  floor  7.8™  long  and  6.2™  wide,  the  strips 
being  84^^™  wide  ? 

19.  A  bin  2.3™  long,  1.1™  wide,  and  1.48™  deep  is  filled 
with  grain.  How  much  is  it  worth  at  ^  2.75  a  hektoliter  ? 
If  the  grain  weighs  .83  times  as  much  as  an  equal  bulk  of 
water,  what  is  the  weight  of  the  contents  in  kilograms  ? 

20.  A  cylindrical  cistern,  25'^™  deep,  holds  636.174°^  of 
water.     Find  the  diameter  of  the  cistern  in  meters. 

21.  How  much  will  it  cost  to  cover  a  floor  5.6™  long,  and 
4.9™  wide,  with  carpeting  75*^™  wide,  at  87  cents  a  meter,  if 
the  strips  run  lengthwise  of  the  room  ?  How  much  if  the 
strips  run  across  the  room  ? 

22.  A  tank  84*^™  deep,  with  a  square  bottom,  contains 
869.4^^  of  sulphuric  acid.  If  the  acid  is  1.84  times  as  heavy 
as  water,  what  is  the  length  of  each  side  of  the  bottom  in 
centimeters  ? 

23.  To  what  depth  must  a  cylindrical  cistern  113<=™  in 
diameter  be  filled,  to  hold  a  metric  ton  of  water  ? 

24.  How  much  will  it  cost  to  paper  a  room  6.4™  long, 
5.4™  wide,  and  3.1™  high,  with  paper  58*^™  wide,  10.7™  to  a 
roll,  at  84  cents  a  roll,  allowing  for  three  doors,  each  85*^™ 
wide  and  2"  high,  two  windows,  each  82*^™  wide  and  17*^™ 
high,  and  a  base-board  3'^'"  wide  ? 

25.  A  cylindrical  tank,  18*^  deep,  contains  2862.783^«  of 
oil.  If  the  oil  is  .9  as  heavy  as  water,  find  the  diameter  of 
the  tank  in  meters. 

257.  Specific  Gravity. 

If  the  specific  gravity  of  any  substance  is  8.7,  a  cubic 
centimeter  of  the  substance  will  weigh  8.7  times  as  much  as 
a  cubic  centimeter  of  water ;  that  is,  it  will  weigh  8.7^. 

A  cubic  decimeter  (or  liter)  of  the  substance  will  weigh 
1000  X  8.7S  or  8.7^«. 


MENSURATION.  189 

A  cubic  meter  of  the  substance  will  weigh  1000  x  8.7^^, 
or  8.7^. 

It  follows  from  the  above  that  the  specific  gravity  of  any 
substance  is : 

1.  The  number  of  grams  in  the  weight  of  a  cubic  centimeter 
of  the  substance. 

2.  The  number  of  kilograms  in  the  weight  of  a  cubic  deci- 
meter {or  liter)  of  the  substance. 

3.  The  number  of  metric  tons  in  the  weight  of  a  cubic  meter 
of  the  substance. 

EXAMPLES. 

1.  Find  the  weight  in  kilograms  of  a  bar  of  aluminum 
(specific  gravity  2.57)  8^™  long,  2*=™  wide,  and  7'"'"  thick. 

The  volume  of  the  bar  is  8<i'"  x  .2dm  x  .07^™^  or  .112cu  dm. 
But  since  the  specific  gravity  of  the  substance  is  2.57,  a  cubic  deci- 
meter of  it  weighs  2.57^^. 

Hence,  the  required  weight  is  .112  x  2.57Kg,  or  .28784Kg,  Ans. 

2.  Find  the  number  of  cubic  centimeters  in  a  piece  of 
silver  (specific  gravity  10.5)  weighing  .2625°°. 

.2625Hg  is  the  same  as  26.25?. 

Since  one  cubic  centimeter  of  silver  weighs  10. 5^,  there  will  be  as 
many  cubic  centimeters  in  26. 25^  as  10.5  is  contained  times  in  26.25. 
Hence,  26.25  ^  10.5  =  2.5c"cm,  Ans. 

3.  If  3™  of  sulphuric  acid  weigh  552^^,  what  is  its  specific 
gravity  ? 

If  3H1  weigh  552Kg,  one  liter  will  weigh  3 -Jo  of  552Kg,  or  1.84^8. 
Hence,  the  specific  gravity  of  the  acid  is  1.84,  Ans. 

4.  Find  the  weight  in  grams  of  5*'"'^"  of  cork  (specific 
gravity  .24). 

5.  Find  the  weight  in  hektograms  of  15^^  of  petroleum 
(specific  gravity  .878). 

6.  How  many  cubic  centimeters  are  there  in  a  piece  of 
lead  (specific  gravity  11.4)  weighing  2131.8^^  ? 


190  ARITHMETIC. 

7.  How  many  hektoliters  of   alcohol    (specific  gravity 
.791)  does  it  take  to  weigh  10203.9"^  ? 

8.  If  8^"°™  of  ice   weigh   7360'^,  what   is   its   specific 
gravity  ? 

9.  If  4.8''^  of  mercury  weigh  65260.8''^,  what  is  its  specific 
gravity  ? 

10.  How  many  metric  tons  of  sand  (specific  gravity  1.42) 
can  be  put  into  a  box  1.5™  long,  19"^'"  wide,  and  57*=™  deep  ? 

11.  Find  the  weight  in  decigrams  of  a  bar  of  platinum 
(specific  gravity  21.5)  5,9*^'"  long,  2.4'=°*  wide,  and  3.8™°^  thick. 

12.  A  block  of  stone  (specific  gravity  2.9),  1.2™  long  and 
8.37'^'"  wide,  weighs  1.893294'^.  Find  its  thickness  in  centi- 
meters. 

13.  A  rod  of  zinc  5™  long,  and  2*=™  in  diameter,  weighs 
11.15268^^.     Find  its  specific  gravity. 

14.  A  brick  2*1'"  long,  62™™  wide,  and  2.4^™  thick,  weighs 
6.19008^2.     pind  its  specific  gravity. 

15.  A  cylindrical  tank,  12*^™  deep,  is  filled  with  linseed 
oil  (specific  gravity  .94).  If  the  weight  of  the  oil  is 
717.604272^^,  what  is  the  diameter  of  the  tank  in  deci- 
meters ? 

16.  Find  the  weight  in  kilograms  of  a  tapering  bar  of 
steel  (specific  gravity  7.8),  5.3™  long,  3*"™  in  diameter  at  one 
end,  and  2*=™  in  diameter  at  the  other. 


RATIO   AND  PROPORTION.  191 

XVI.    RATIO    AND    PROPORTION. 

258.  The  Ratio  of  one  number  to  another  is  the  quotient 
obtained  by  dividing  the  first  number  by  the  second. 

Thus,  tlie  ratio  of  3  to  5  is  f ;  it  is  also  expressed  3 : 5. 

259.  The  first  term  of  a  ratio  is  called  the  antecedent, 
and  the  second  term  the  consequent. 

Thus,  in  the  ratio  3:5,  3  is  the  antecedent,  and  5  the 
consequent. 

260.  The  ratio  of  two  quantities  of  the  same  kind,  when 
expressed  in  terms  of  the  same  unit,  is  the  ratio  of  the  num- 
bers by  which  they  are  expressed. 

Thus,  the  ratio  of  f  10  to  $  7  is  10 :  7. 

The  ratio  of  3  bushels  to  19  pecks  is  the  ratio  of  12  pecks 
to  19  pecks,  or  12 :  19. 

Note.  No  ratio  can  exist  between  quantities  which  are  not  of  the 
same  kind,  such  as  feet  and  pounds. 

261.  A  ratio  being  an  expression  of  division,  it  follows 
from  Arts.  87  and  93  that 

Both  terms  of  a  ratio  may  he  multiplied  or  divided  by  the 
same  number,  ivithout  altering  the  value  of  the  ratio. 

Thus,  the  ratio  21 :  35,  by  dividing  each  term  by  7,  is 
equal  to  the  ratio  3 :  5. 

262.  A  ratio  is  said  to  be  compounded  of  two  or  more 
others  when  its  antecedent  is  the  product  of  their  antece- 
dents, and  its  consequent  the  product  of  their  consequents. 

3-5) 

Thus,  the  ratio  compounded  of     "     [-is  3x7:5x4,  or 

21 :  20.  ^  *  ^  ^ 

EXAMPLES. 

263.  Simplify  the  following  ratios  : 

1.  98:245.  3.    $  7.13  :$  11.47.  5.    1089:1573. 

2.  i^:«.  4.   4i:4A.  6.    3^^ :  2^. 


192  ARITHMETIC. 

7.  What  is  the  ratio  of  25  lb.  7  oz.  av.  to  32  lb.  6  oz.  av.  ? 

8.  What  is  the  ratio  of  £  5  6s.  3d.  to  £  3  3s.  9d.  ? 

9.  What  is  the  ratio  of  3  rd.  5  yd.  1  ft.  5  in.  to  5  rd. 
0  yd.  2  ft.  3  ill.  ? 

10.   Which  is  the  greater,  12  :  13  or  25  :  27  ? 

Find  the  ratios  compounded  of : 

2:3)  8:9   -)  27:25)  68:57 

11.      6:7  I-     12.    11 :  16  [  .     13.    28  :  45  [-  •     14.  76  :  69 

14:5)  12: 11 3  20:21)  92:85 

PROPORTION. 

264.  A  Proportion  is  an  equality  of  ratios. 

Thus,  the  statement  that  the  ratio  of  3  to  5  is  equal  to 
the  ratio  of  6  to  10  forms  a  proportion,  which  may  be  written 
in  the  forms 

3  :  5  =  6 :  10,  f  =  A,  or  3  :  5  :  :  6  :  10. 

265.  The  first  and  fourth  terms  of  a  proportion  are  called 
the  extremes,  and  the  second  and  third  terms  the  means. 

Thus,  in  the  proportion  3  :  5  =  6  :  10,  3  and  10  are  the 
extremes,  and  5  and  6  the  means. 

266.  In  a  proportion  in  which  the  means  are  equal,  either 
mean  is  called  a  Mean  Proportional  between  the  first  and 
fourth  terms,  and  the  fourth  term  is  called  a  Third  Propor- 
tional to  the  first  and  second  terms. 

Thus,  in  the  proportion  4:6  =  6:9,  6  is  a  mean  propor- 
tional between  4  and  9,  and  9  is  a  third  proportional  to  4 
and  6. 

A  Fourth  Proportional  to  three  numbers  is  the  fourth 
term  of  a  proportion,  whose  first  three  terms  are  the  three 
numbers  in  their  order. 

Thus,  in  the  proportion  3  :  5  =  6  :  10,  10  is  a  fourth  pro- 
portional to  3,  5,  and  6. 


RATIO   AND   PROPORTION.  193 

PROPERTIES    OF    PROPORTIONS. 

267.  Let  us  consider  the  proportion 

3:5  =  6:10,  or  i  =  ^. 

If  each  of  these  equal  fractions  be  multiplied  by  the 
product  of  the  two  denominators,  5  x  10,  the  products  will 
evidently  be  equal ;  that  is, 

5  X  10  X  f  =  5  X  10  X  T^. 
Cancelling  5  in  the  first  product,  and  10  in  the  second, 
we  have 

10x3  =  5x6.  (1) 

Hence,  in  any  proportion,  the  product  of  the  extremes  is 
equal  to  the  product  of  the  means. 

268.  If  each  of  the  equal  products  in  (1),  Art.  267,  be 
divided  by  3,  the  quotients  will  evidently  be  equal. 

That  is,        122<3  =  5x6.^^,0  =  5|i. 

Hence,  in  any  proportion,  either  extreme  is  equal  to  the 
product  of  the  means  divided  by  the  other  extreme. 

In  like  manner,  either  mean  is  equal  to  the  product  of  the 
extremes  divided  by  the  other  mean. 


It  follows  from  Art.  267  that  the  square  of  the 
mean  proportional  between  two  numbers  is  equal  to  the 
product  of  the  numbers. 

Hence,  the  mean  proportional  between  two  numbers  is  equal 
to  the  square  root  of  their  product. 

EXAMPLES. 

270.   1.  Find  a  fourth  proportional  to  7,  10,  and  21. 
By  Art.  268,  the  required  fourth  proportional  is 

^^  ^  ^^,  or  30,  Ans. 


194  ARITHMETIC. 

2.  Find  a  mean  proportional  between  |  and  ^|. 
By  Art.  269,  the  required  mean  proportional  is 

3.  Find  a  fourth  proportional  to  65,  80,  and  91. 

4.  Find  a  third  proportional  to  25  and  30. 

5.  Find  a  mean  proportional  between  12  and  48. 

6.  Find  a  third  proportional  to  J  and  f. 

7.  Find  a  fourth  proportional  to  fj,  f,  and  ^. 

8.  Find  a  mean  proportional  between  5|-  and  18  j^. 

9.  What  is  the  second  term  of  a  proportion  whose  first, 
third,  and  fourth  terms  are  5f ,  4|-,  and  If  ? 

10.   Find  a  fourth  proportional  to  ff,  -fl,  and  -i^. 

PROBLEMS    IN    PROPORTION. 

271.  1.  If  35  yards  of  cloth  cost  $  78.75,  how  much  will 
47  yards  cost  ? 

It  is  convenient  to  arrange  the  proportion  so  that  the 
required  answer  shall  be  the  fourth  term,  and  the  quantity 
which  is  of  the  same  kind  as  the  answer  the  third  term. 

In  the  present  case,  since  cost  is  required,  the  third  term 
will  be  ^78.75. 

Let  the  fourth  term  be  represented  by  x. 

The  other  two  quantities,  35  yards  and  47  yards,  are  taken 
to  form  the  first  ratio ;  and  since  47  yards  will  evidently 
cost  more  than  35  yards,  the  first  term  of  the  proportion 
should  be  35  yards,  and  the  second  term  47  yards. 

The  proportion  will  then  stand  as  follows  : 

35  yards  :  47  yards  =  ^  78.75 :  x. 

The  value  of  x  may  be  obtained  by  means  of  the  principle 
of  Art.  268 ;  that  is,  by  dividing  the  product  of  the  means 
by  the  other  extreme. 


RATIO   AND   PROPORTION.  195 

In  order  to  avoid  the  multiplication  of  yards  by  dollars, 
we  may  consider  the  terms  of  each  ratio  replaced  by  the 
numbers  which  express  how  many  times  they  contain  the 
unit. 

Thus,  35  :  47  =  78.75  :  x. 

Whence,  x  =  ^^  ^  l^'^^  =  105.75. 

35 

Then  the  required  cost  is  $105.75,  Ans. 

2.  If  27  men  can  do  a  piece  of  work  in  15  days,  how  many 
days  will  it  take  36  men  to  do  it  ? 

Since  the  answer  is  to  be  days,  we  make  15  days  the 
third  term. 

Now  36  men  will  evidently  require  less  time  than  27  men 
to  perform  the  work ;  we  therefore  make  36  men  the  first 
term,  and  27  men  the  second  term. 

Then,  omitting  reference  to  the  units, 

36 :  27  =  15  :  x. 
Or,  .  =  M^|  =  1H. 

Then  the  required  time  is  11^  days,  Ans. 

From  the  above  examples,  we  derive  the  following 

RULE. 

Make  the  quantity  which  is  of  the  same  kind  as  the  answer  . 
the  third  term. 

If,  from  the  nature  of  the  problem,  the  answer  is  to  be  greater 
than  the  third  term,  make  the  smaller  of  the  remaining  quan-l 
tities  the  first,  and  the  greater  the  second  term;  but  if  thel 
answer  is  to  be  less  than  the  third  term,  make  the  greater  of  thi 
quantities  the  first  term,  and  the  smaller  the  second  term. 

Finally,  divide  the  product  of  the  means  by  the  giver 
extreme. 


196  ARITHMETIC. 

EXAMPLES. 

3.  If  23  gallons  of  molasses  cost  $  4.37,  how  much,  will 
29  gallons  cost  ? 

4.  If  51  bushels  of  grain  cost  $  21.42,  how  much  will  38 
bushels  cost? 

5.  If  41  men  can  do  a  piece  of  work  in  35  days,  how  many 
days  will  it  take  28  men  to  do  it  ? 

6.  If  37  men  can  do  a  piece  of  work  in  63  days,  how 
many  days  will  it  take  42  men  to  do  it  ? 

7.  If  a  man  travels  705  miles  in  25  days,  how  many 
miles  will  he  travel  in  23  days  ? 

8.  If  a  certain  amount  of  provisions  will  last  50  men 
121  days,  how  many  days  will  it  last  77  men  ? 

9.  If  4|- bushels  of  oats  cost  $2.87i  how  much  will  7-| 
bushels  cost  ? 

10.  If  a  man  can  perform  a  certain  journey  in  84  hours, 
travelling  9|-  miles  an  hour,  how  many  hours  will  it  take 
him  travelling  16|-  miles  an  hour? 

11.  A  bankrupt  pays  76  cents  on  a  dollar.  If  a  certain 
creditor  receives  $  251.75,  what  was  his  original  claim  ? 

12.  If  a  field  26  rods  long  and  10  rods  wide  be  worth 
$  650,  how  much  will  a  field  25  rods  long  and  18  rods  wide 
be  worth  ? 

13.  If  234  men  can  do  a  piece  of  work  in  11|  hours,  how 
many  men  will  it  take  to  do  it  in  9|  hours  ? 

14.  If  5i  yards  of  cloth  cost  $  8.25,  how  much  will  13f 
yards  cost  ? 

15.  If  a  man  5  ft.  9f  in.  in  height  casts  a  shadow  3  ft. 
2f  in.  in  length,  what  is  the  height  of  a  steeple  that  casts  a 
shadow  88  ft.  6i  in.  in  length? 

16.  If  8^  tons  of  coal  cost  ^36|,  how  many  tons  can  be 
bought  for  1 39^? 


RATIONAL)  MOPORTION.  197 


17.  If  a  train  travels  6468  yards  in  5|-  minutes,  what  is 
its  rate  in  miles  an  hour  ?  , 

18.  If  £2  8s.  Sd.  be  worth  ^11.92,  how  much  is  £6  Is. 
8d.  worth  ? 

19.  If  a  horse  travels  65  rods  in  104  seconds,  how  many- 
minutes  will  it  take  him  to  travel  a  mile  ? 

20.  A  pipe  which  discharges  3  qt.  1  pt.  2  gi.  in  one  second, 
empties  a  tank  in  5  min.  57  sec.  How  long  will  it  take  a 
pipe  which  discharges  2  qt.  0  pt.  1  gi.  in  one  second  to  empty 
the  tank  ? 

21.  If  a  train  performs  a  certain  journey  in  2-|  hours, 
travelling  at  the  rate  of  2750  feet  a  minute,  how  long  will  it 
take  it  travelling  at  the  rate  of  18|-  yards  a  second  ? 

22.  A  garrison  of  357  men  had  food  for  112  days ;  but 
some  reinforcements  having  been  received,  the  food  lasted 
only  98  days.  How  many  men  were  received  as  reinforce- 
ments ? 

23.  If  3  cu.  ft.  1008  cu.  in.  of  water  weigh  223  lb.  15^  oz., 
how  much  will  4  cu.  ft.  720  cu.  in.  of  water  weigh  ? 

24.  A  piece  of  work  was  to  have  been  done  by  45  men  in 
83  days  ;  but  after  27  days,  21  men  were  sent  away.  How 
long  did  it  take  the  remaining  24  men  to  complete  the 
work  ? 

25.  If  a  field  10  rd.  3  yd.  long  and  7  rd.  4  yd.  wide  be 
worth  $  147^,  how  much  will  a  field  9  rd.  3  yd.  long  and 
8  rd.  4  yd.  wide  be  worth  ? 

COMPOUND  PROPORTION. 

272.  A  Compound  Proportion  is  a  proportion  one  of  whose 
ratios  is  a  compound,  ratio  (Art.  262)  ;  as, 

3 


;n 


.       42:40, 

7 


198  ARITHMETIC. 


PROBLEMS. 


273.  1.  If  4  men  can  build  64  feet  of  wall  in  8  days  of  9 
hours  each,  how  many  men  will  it  take  to  build  80  feet  of 
wall  in  3  days  of  10  hours  each  ? 

Since  the  answer  is  to  be  men,  we  make  4  men  the 
third  term. 

If  the  answer  depended  only  on  the  number  of  feet  built, 
it  would  be  greater  than  the  third  term,  for  it  will  take 
longer  to  build  80  feet  than  64  feet ;  hence  the  first  ratio  is 
64 :  80. 

Again,  if  the  answer  depended  only  on  the  number  of 
days  worked,  it  would  be  greater  than  the  third  term,  for  it 
will  take  more  men  to  build  the  wall  in  3  days  than  in  8 
days  ;  hence  the  second  ratio  is  3  :  8. 

Also,  if  the  answer  depended  only  on  the  number  of  hours 
worked  each  day,  it  would  be  less  than  the  third  term,  for 
it  will  take  fewer  men  to  build  the  wall  working  10  hours 
than  working  9  hours  a  day ;  hence  the  third  ratio  is  10  :  9. 

The  work  now  stands  as  follows  : 

64:80) 

3:8    [-  =  4:  a;.  ' 

10:9    ) 

g  3 

Whence,    a:  =  ^xJ0xl_X^  ^  ^2  men,  ^ns. 

2.  If  10  men  can  build  108  feet  of  wall  in  6  days,  how 
many  feet  can  15  men  build  in  5  days  ? 

3.  If  22  men  can  do  a  piece  of  work  in  3  days  of  9  hours 
each,  how  many  days  of  11  hours  each  will  it  take  15  men 
to  do  the  same  work  ? 

4.  If  18  men  can  build  57  rods  of  ditch  in  26  days,  how 
many  men  will  it  take  to  build  38  rods  in  39  days  ? 


RATIO   AND   PROPORTIOX.  I99 

5.  If  16  men  can  do  a  piece  of  work  in  9  days  of  8|  hours 
each,  how  many  men  will  it  take  to  do  the  same  work  in  6 
days  of  10  hours  each  ? 

6.  If  13  horses  consume  65  bushels  of  oats  in  24  days, 
how  many  bushels  will  7  horses  consume  in  32  days  ? 

7.  If  5  men  can  build  200  feet  of  fence  in  4  days,  in  how 
many  days  can  7  men  build  350  feet  ? 

8.  If  a  pasture  of  13  acres  will  feed  9  cows  for  4 J  months, 
how  many  cows  will  21  acres  feed  for  5^1:  months  ? 

9.  If  a  man  can  travel  156  miles  in  6  days  of  8J  hours 
each,  how  many  days  of  12J  hours  each  will  it  take  him  to 
travel  85  miles  ? 

10.  If  6  men  can  do  a  piece  of  work  in  9  days  of  10  hours 
each,  in  how  many  hours  a  day  can  12  men  do  the  same 
work  in  5  days  ? 

11.  If  a  man  can  travel  125  miles  in  5  days  of  6f  hours 
each,  how  far  can  he  travel  in  8  days  of  7-^  hours  each  ? 

12.  If  8  horses  consume  4|-  tons  of  hay  in  32  days,  how 
many  days  will  6|  tons  last  9  horses  ? 

13.  If  a  10-cent  loaf  of  bread  weighs  7J  ounces,  when 
wheat  is  $  lyig-  a  bushel,  what  ought  a  5-cent  loaf  to  weigh 
when  wheat  is  $  1 J  a  bushel  ? 

14.  If  a  man  can  travel  63  miles  in  3  days  of  7  hours 
each,  how  many  hours  a  day  must  he  travel  to  cover  105 
miles  in  4  days  ? 

15.  If  6  men  can  build  102  yards  of  wall  in  17  days  of  7 
hours  each,  how  many  yards  can  7  men  build  in  8  days  of 
9  hours  each  ? 

16.  If  a  piece  of  metal  7  feet  long,  4i  inches  wide,  and 
5  inches  thick,  weighs  550  pounds,  how  much  will  a  piece 
of  the  same  metal  weigh  that  is  12  feet  long,  5J  inches  wide, 
and  3  inches  thick  ? 


200  ARITHMETIC. 

17.  If  7  men  can  build  140  feet  of  fence  in  3  days  of  8 J 
hours  each,  in  how  many  hours  a  day  can  6  men  build  240 
feet  of  fence  in  5  days  ? 

18.  If  a  bin  8  feet  long,  4J  feet  wide,  and  2^  feet  deep, 
holds  67J  bushels,  how  wide  must  a  bin  be  made  that  is 
20  feet  long  and  4i  feet  deep,  to  hold  450  bushels  ? 

19.  If  13  men  can  build  39  rods  of  ditch  in  4  days  of  10^- 
hours  each,  how  many  days  of  9^  hours  each  will  it  take  11 
men  to  build  55  rods  ? 

20.  If  a  piece  of  iron  18  inches  long,  8  inches  wide,  and 
li  inches  thick,  weighs  54  pounds,  how  much  will  a  piece 
weigh  that  is  15  inches  long,  12  inches  wide,  and  If  inches 
thick  ? 

21.  If  a  tank  7  feet  long,  4  feet  wide,  and  2^  feet  deep, 
holds  8^  hogsheads  of  water,  how  many  hogsheads  will  a 
tank  hold  that  is  6  feet  long,  5  feet  wide,  and  S^  feet  deep  ? 

22.  If  a  piece  of  stone  6  feet  long,  1^-  feet  wide,  and  2 
inches  thick,  weighs  240  pounds,  how  long  must  a  piece  of 
the  same  stone  be  that  is  2^  feet  wide,  and  3  inches  thick, 
to  weigh  840  pounds  ? 

23.  If  3  men  can  do  -^^  of  a  certain  piece  of  work  in  5 
days  of  12  hours  each,  how  many  men  will  it  take  to  do  -J  of 
the  work  in  6  days  of  8  hours  each  ? 

24.  If  a  tank  6  feet  long,  3  feet  wide,  and  2  feet  deep, 
holds  4i  hogsheads  of  water,  how  deep  must  a  tank  be  that 
is  8  feet  long,  and  5 J  feet  wide,  to  contain  22|-  hogsheads  of 
water  ? 

25.  If  16  men  can  do  f  of  a  certain  piece  of  work  in 
20  days  of  7  hours  each,  in  how  many  days  of  8  hours  each 
can  12  men  do  f  of  the  work  ? 

26.  If  12  blocks  of  stone,  each  3  feet  long,  1^  feet  wide, 
and  6  inches  thick,  weigh  together  1035  pounds,  how  much 
will  18  blocks  of  the  same  stone  weigh,  each  2^  feet  long, 
1  foot  wide,  and  8  inches  thick  ? 


RATIO   AND   PROPORTION.  201 

27.  If  14  men  can  build  a  wall  80  feet  long,  4  feet  wide, 
and  6  feet  high,  in  24  days,  how  many  days  will  it  take 
18  men  to  build  a  wall  72  feet  long,  5  feet  wide,  and  8  feet 
high? 

28.  If  45  men  can  dig  a  ditch  100  feet  long,  20  feet  wide, 
and  8  feet  deep,  in  5  days,  how  many  men  will  it  take  to 
dig  a  ditch  120  feet  long,  32  feet  wide,  and  9  feet  deep,  in  6 
days  ? 

29.  'If  6  men  can  build  a  wall  72  feet  long,  5  feet  wide, 
and  8  feet  high,  in  8  days  of  10  hours  each,  how  many  men 
will  it  take  to  build  a  wall  60  feet  long,  4  feet  wide,  and 
9  feet  high,  in  5  days  of  9  hours  each  ? 

30.  If  24  men  can  dig  a  trench  96  feet  long,  10  feet  wide, 
and  6  feet  deep,  in  6  days  of  8  hours  each,  how  many  days 
of  11  hours  each  will  it  take  21  men  to  dig  a  trench  80  feet 
long,  11  feet  wide,  and  7  feet  deep  ? 

PARTITIVE  PROPORTION. 

274.  Partitive  Proportion  is  the  process  of  dividing  a 
number  into  parts  proportional  to  certain  given  numbers. 

PROBLEMS. 

275.  1.  Divide  98  into  parts  proportional  to  2,  3,  4,  and  5. 
The  first  part  is  to  be  -f,  the  second  part  f,  and  the  third 

part  f ,  of  the  fourth  part. 

Whence,  the  sum  of  all  the  parts  must  be  f  +  f  +  f  +  f ? 
or  -1^  of  the  fourth  part. 

But  the  sum  of  all  the  parts  is  98. 

Whence,  the  fourth  part  must  be  y^^  of  98,  or  35 ;  and  the 
first,  second,  and  third  parts  are,  respectively,  |-,  f,  and  |  of 
35,  or  14,  21,  and  28,  Ans. 

In  the  above  example,  the  required  parts  are,  respectively, 
-^,  A,  ^,  and  ^  of  98. 


202  ARITHMETIC. 

In  any  similar  case,  we  form  fractions  having  the  given 
numbers  for  their  numerators,  and  the  sum  of  these  numbers 
for  their  common  denominator. 

2.  Divide  54  into  parts  proportional  to  -|,  f ,  and  -J. 
Eeducing  the  fractions  to  their  least  common  denominator, 

this  is  the  same  as  dividing  54  into  parts  proportional  to 
tVj  A'  ^^^  li'j  ^^y  ^^^0  parts  proportional  to  8,  9,  and  10. 

8  +  9+10  =  27. 

Whence,  the  required  parts  are,  respectively,  -^  of  54, 
^  of  54,  and  ^^  of  54 ;  or,  16,  18,  and  20,  Ans. 

3.  Divide  105  into  parts  proportional  to  2,  5,  and  8. 

4.  Divide  324  into  parts  proportional  to  3^  and  4-|, 

5.  Divide  455  into  parts  proportional  to  3,  5,  7,  9,  and  11. 

6.  Divide  2466  into  parts  proportional  to  li,  IJ,  li, 
and  l^-. 

7.  Divide  406  into  two  parts,  the  second  of  which  shall 
be  seven  times  the  first. 

8.  An  alloy  contains  30  parts  copper,  18  parts  zinc,  and 
7  parts  tin.  How  many  pounds  of  each  metal  will  be  re- 
quired to  make  935  pounds  of  the  alloy  ? 

9.  Three  men,  A,  B,  and  C,  invested  money  in  a  certain 
enterprise;  A  putting  in  $1250,  B  $1600,  and  C  $2040. 
If  they  gain  $  1467,  how  shall  it  be  divided  ? 

10.  A  certain  explosive  mixture  contains  16  parts  salt- 
petre, 3  parts  charcoal,  and  5  parts  sulphur.  How  many 
pounds  of  each  substance  will  be  required  to  make  274 
pounds  of  the  mixture  ? 

11.  Divide  1386  into  four  parts,  such  that  the  second  part 
shall  be  twice  the  first,  the  third  part  three  times  the  second, 
and  the  fourth  part  four  times  the  third. 

12.  A  man  divided  the  sum  of  $  163.52  between  his  four 
children,  in  proportion  to  the  numbers  8,  12,  15,  and  21. 
How  much  did  each  receive  ? 


RATIO   AND  PROPORTION.  203 

13.  Air  contains  21  parts  oxygen  to  79  parts  nitrogen. 
How  many  cubic  feet  of  each  gas  are  there  in  a  receptacle 
5  ft.  4  in.  long,  5  ft.  3  in.  wide,  and  4  ft.  6  in.  deep  ? 

14.  Divide  66  into  three  parts,  snch  that  the  second  part 
shall  be  f  of  the  lirst,  and  the  third  part  ^  of  the  second. 

15.  A  certain  sum  of  money  was  divided  between  three 
men,  in  proportion  to  the  numbers  41,  32,  and  19.  If  the 
first  man  received  $  51.25,  what  was  the  amount  divided  ? 

16.  A  certain  number  is  divided  into  parts  proportional 
to  the  numbers  3,  6, 11,  and  18,  such  that  the  fourth  part 
exceeds  the  third  by  35.     What  are  the  parts  ? 

SIMILAR  SURFACES  AND  SOLIDS. 

276.  Similar  surfaces  or  solids  are  those  which  have  the 
same  form. 

Thus,  two  circles  of  unequal  diameters  are  similar  surfaces. 

It  is  proved  in  Geometry  that : 

1.  Corresponding  lines  in  similar  surfaces  or  similar  solids 
are  proportional. 

Thus,  in  two  circles,  the  circumference  of  the  first  is  to 
the  circumference  of  the  second  as  the  diameter  of  the  first 
is  to  the  diameter  of  the  second. 

2.  The  areas  of  similar  surfaces  are  to  each  other  as  the 
squares  of  their  corresponding  lines. 

Thus,  the  area  of  one  circle  is  to  the  area  of  another  as  the 
square  of  the  radius  of  the  first  is  to  the  square  of  the  radius 
of  the  second. 

3.  The  volumes  of  similar  solids  are  to  each  other  as  the 
cubes  of  their  corresponding  lines. 

Thus,  the  volume  of  one  sphere  is  to  the  volume  of  another 
as  the  cube  of  the  diameter  of  the  first  is  to  the  cube  of  the 
diameter  of  the  second. 


204  ARITHMETIC. 

EXAMPLES. 

1.  If  the  volume  of  a  pyramid  whose  altitude  is  7  in. 
is  686  cu.  in.,  what  is  the  volume  of  a  similar  pyramid  whose 
altitude  is  12  inches  ? 

Let  X  represent  the  volume  of  the  second  pyramid. 
Then,  since  the  volumes  of  similar  solids  are  to  each  other  as  the 
cubes  of  their  corresponding  lines,  we  have 
686 :  a;  =  73 :  123. 

Whence,  x  =  ^^^^Jl",  or  3456  cu.  in.,  Ans. 

2.  If  the  area  of  a  rectangle  whose  base  is  12  ft.  is  96  sq. 
ft.,  what  is  the  base  of  a  similar  rectangle  whose  area  is 
216  sq.  ft.  ? 

Let  X  represent  the  base  of  the  second  rectangle. 
Then,  since  the  areas  of  similar  surfaces  are  to  each  other  as  the 
squares  of  their  corresponding  lines,  we  have 
96  :  216  =  122  .  ^2. 

Whence,  x^  =  ^^^  ^  ^^^  =  324. 

'  96 

Therefore,  x  =  V324,  or  18  ft.,  Ans. 

3.  If  the  area  of  a  parallelogram  whose  base  is  15  in.  is 
375  sq.  in.,  what  is  the  area  of  a  similar  parallelogram  whose 
base  is  18  in.  ? 

4.  If  the  area  of  a  triangle  whose  altitude  is  12  in.  is 
72  sq.  in.,  what  is  the  altitude  of  a  similar  triangle  whose 
area  is  32  sq.  in.  ? 

5.  If  the  volume  of  a  frustum  of  a  cone  whose  altitude 
is  16  in.  is  360  cu.  in.,  what  is  the  volume  of  a  similar  frus- 
tum of  a  cone  whose  altitude  is  20  in.  ? 

6.  If  the  volume  of  a  prism  whose  altitude  is  9  ft.  is 
171  cu.  ft.,  what  is  the  altitude  of  a  similar  prism  whose 
volume  is  50|  cu.  ft.  ? 

7.  If  the  area  of  a  trapezoid  whose  altitude  is  1  ft.  10  in. 
is  4  sq.  ft.  104  sq.  in.,  what  is  the  altitude  of  a  similar 
trapezoid  whose  area  is  2  sq.  ft.  94J  sq.  in.  ? 


•    RATIO   AND   PROPORTION.  205 

8.  If  the  volume  of  a  cylinder  the  radius  of  whose  base  is 
5  in.  is  625  cu.  in.,  what  is  the  radius  of  the  base  of  a  similar 
cylinder,  whose  volume  is  1715  cu.  in.  ? 

9.  Two  bins  of  the  same  form  contain,  respectively,  375 
and  648  bushels  of  wheat.  If  the  first  bin  is  3  ft.  9  in.  deep, 
what  is  the  depth  of  the  second  ? 

10.  If  a  circular  plate  of  silver  whose  diameter  is  8  inches 
is  worth  $  76.80,  what  is  the  diameter  of  a  similar  plate 
whose  value  is  $97.20  ? 

11.  If  it  costs  $  26.25  to  concrete  the  floor  of  a  cellar 
whose  length  is  35  ft.,  how  much  will  it  cost  to  concrete  the 
floor  of  a  similar  cellar  whose  length  is  42  ft.  ? 

12.  If  a  sphere  6  inches  in  diameter  weighs  47  lb.  4  oz., 
what  is  the  weight  of  a  sphere  of  the  same  material  whose 
diameter  is  10  inches  ? 

13.  A  cone  whose  altitude  is  10  in.  weighs  24  lb.  At 
what  distance  from  the  vertex  must  it  be  cut  by  a  plane 
parallel  to  the  base,  so  that  the  frustum  cut  off  may  weigh 
12  lb.  ? 

14.  If  a  pipe  whose  diameter  is  2f  in.  can  empty  a  tank 
in  3  hr.  44  min.,  how  long  will  it  take  a  pipe  whose  diameter 
is  3J  in.  to  empty  it  ? 

15.  If  a  sphere  whose  diameter  is  2  ft.  1  in.  weighs  3125 
lb.,  what  is  the  diameter  of  a  sphere  of  the  same  material 
whose  weight  is  819.2  lb.  ? 


206  ARITHMETIC. 


XVII.    PARTNERSHIP. 

277.  A  Partnership  is  an  association  formed  by  two  or 
more  persons  for  the. transaction  of  business. 

The  persons  forming  a  partnership  are  csilled  partners. 

278.  The  Capital  or  Stock  is  the  money  or  property  which 
the  partners  invest  in  the  business. 

The  Assets  or  Resources  of  a  partnership  comprise  all 
property  of  whatever  nature  belonging  to  it. 
The  Liabilities  are  its  debts. 

SIMPLE  PARTNERSHIP. 

279.  A  Simple  Partnership  is  one  in  which  the  capital  of 
each  partner  is  invested  for  the  same  length  of  time. 

In  such  a  case,  the  profits  or  losses  are  shared  by  the  part- 
ners in  proportion  to  the  amounts  which  they  have  invested. 

PROBLEMS. 

280.  1.  A,  B,  and  C  form  a  partnership;  A  putting  in 
$3500,  B  $3200,  and  C  $2800.  If  $5700  is  gained,  what 
is  each  partner's  share  of  the  profits  ? 

The  profits  are  shared  by  A,  B,  and  C  in  proportion  to  the  numbers 
3500,  3200,  and  2800  ;  or,  dividing  each  number  by  100,  in  proportion 
to  the  numbers  35,  32,  and  28. 

35  +  32  +  28  =  95  ;  whence,  by  Art.  275, 

A's  share  is  f  f  of  $  5700,  or  f  2100  ; 
B's  share  is  |f  of  $  5700,  or  $  1920  ; 
C's  share  is  ff  of  $  5/00,  or  $  1680,  Ans. 

Note.  In  any  case  like  the  above,  when  the  numbers  in  proportion 
to  which  the  profits  are  shared  have  a  common  factor,  they  may  be 
divided  by  this  factor  before  applying  the  rule. 

2.  A  and  B  form  a  partnership;  A  putting  in  $3750, 
and  B  $  5375.  They  gained  $  1460.  What  is  each  partner's 
share  of  the  profits  ? 


PARTNERSHIP.  207 

3.  Allen,  Brown,  and  Cole  enter  into  partnership ;  Allen 
putting  in  $9000,  Brown  $10800,  and  Cole  $5760.  They 
gain  $  3550.     What  is  each  partner's  share  of  the  gain  ? 

4.  A  and  B  formed  a  partnership;  A  putting  in  three 
times  as  much  capital  as  B.  They  lost  $  257.  What  was 
each  partner's  share  of  the  loss  ? 

5.  A  bankrupt  owes  to  A  $  132 ;  to  B  $  165 ;  and  to  C 
$  198.  If  his  entire  resources  are  $  184.65,  what  is  the 
share  of  each  creditor  ? 

6.  A,  B,  and  C  entered  into  partnership ;  A  putting  in 
twice  as  much  capital  as  B,  and  B  putting  in  five  times  as 
much  as  C.  They  lost  $  27392.  What  was  each  partner's 
share  of  the  loss  ? 

7.  Hale  and  Hunt  formed  a  partnership ;  Hale  putting 
in  f  as  much  capital  as  Hunt.  They  gained  $  3795.  What 
was  each  partner's  share  of  the  profits  ? 

8.  A,  B,  and  C  formed  a  partnership ;  B  putting  in  f  as 
much  capital  as  A,  and  C  f  as  much  as  B.  They  gained 
$  6080.43.     What  was  each  partner's  share  of  the  profits  ? 

9.  Four  men.  A,  B,  C,  and  D,  hired  a  pasture  for  $  19.50 
a  month.  A  put  in  7  horses,  B  5  horses,  C  8  horses,  and  D 
6  horses.     What  was  each  man's  share  of  the  rent  ? 

10.  A,  B,  and  C  enter  into  partnership ;  A  putting  in  f 
as  much  capital  as  B,  and  -f-  as  much  as  C.  They  lost 
$2531.20.     What  was  each  partner's  share  of  the  loss  ? 

COMPOUND  PARTNERSHIP. 

281.  A  Compound  Partnership  is  one  in  which  the  amounts 
invested  by  the  partners  are  employed  for  unequal  times. 

PROBLEMS. 

282.  1.  A,  B,  and  C  entered  into  partnership.  A  put  in 
$  300  for  7  months,  B  $  400  for  6  months,  and  C  $  270  for 
10  months.  They  gained  $  216.  What  was  each  partner's 
share  of  the  profits  ? 


208  ARITHMETIC. 

Investing  $300  for  7  months  is  the  same  as  investing  $2100  for 
1  month ;  $  400  for  6  months  is  tlie  same  as  $  2400  for  1  month ;  and 
$  270  for  10  months  is  the  same  as  $  2700  for  1  month. 

Then  the  profits  are  sliared  by  the  partners  in  proportion  to  the 
numbers  2100,  2400,  and  2700  ;  or,  dividing  each  number  by  300,  in 
proportion  to  the  numbers  7,  8,  and  9. 
7  +  8  +  9  =  24  ;  whence, 

A's  share  is  /^  of  $216,  or  |63 ; 
B's  share  is  ^-^  of  $216,  or  $  72  ; 
C's  sliare  is  -^-^  of  $216,  or  $81,  Ans. 

From  the  above  example,  we  derive  the  following 

RULE. 

Multiply  the  capital  of  each  partner  by  the  time  which  it 
is  employed,  and  divide  the  profits  or  losses  in  proportion  to 
the  products. 

2.  A  and  B  formed  a  partnership  ;  A  putting  in  $  840 
for  5  months,  and  B  $720  for  7  months.  They  gained 
$  356.40.     What  was  each  partner^s  share  of  the  profits  ? 

3.  A,  B,  and  C  entered  into  partnership.  A  put  in  $  825 
for  4  months,  B  $  900  for  11  months,  and  C  $  687.50  for  8 
months.  They  lost  $467.50.  What  was  each  partner's 
share  of  the  loss  ? 

4.  A,  B,  C,  and  D  hired  a  pasture  for  $  116.60.  A  put 
in  4  cows  for  3  months,  B  one  cow  for  6  months,  C  3  cows 
for  5  months,  and  D  5  cows  for  4  months.  What  was  each 
man's  share  of  the  rent  ? 

5.  Adams  and  Burke  entered  into  partnership.  Adams 
put  in  $3500  for  9  months,  and  Burke  $4900  for  6  months. 
They  lost  $2320.  What  was  each  partner's  share  of  the 
loss? 

6.  A,  B,  and  C  agreed  to  do  a  certain  piece  of  work  for 
$  3655.  A  furnished  25  men  for  18  days,  B  18  men  for  20 
days,  and  C  32  men  for  15  days.  What  was  each  man's 
share  of  the  amount? 


PARTNERSHIP.  209 

7.  Rand,  Sears,  and  Thomas  formed  a  partnership. 
Rand  put  in  $  5200  for  11  months.  Sears  $4500  for  13 
months,  and  Thomas  $3900  for  16  months.  They  gained 
$  2192.     What  was  each  partner's  share  of  the  profits  ? 

8.  A  and  B  entered  into  partnership  for  one  year.  A 
put  in  $  500  for  5  months,  and  $  500  more  for  the  remainder 
of  the  year;  B  put  in  $600  for  8  months,  and  $200  more 
for  the  remainder  of  the  year.  They  gained  $  980.  What 
was  each  partner's  share  of  the  profits  ? 

9.  A,  B,  and  C  hired  a  pasture  for  $  110.50.  A  put  in 
20  cows  for  5  months,  B  16  horses  for  3  months,  and  C  42 
sheep  for  4  months.  If  5  cows  be  considered  as  equivalent 
to  3  horses,  and  6  sheep  to  one  horse,  how  much  should 
each  man  pay  ? 

10.  Fuller  and  Gray  entered  into  partnership  for  one 
year.  Fuller  put  in  $2300,  and  at  tfie  end  of  8  months 
took  out  $900;  Gray  put  in  $3100,  and  at  the  end  of  3 
months  took  out  $  800,  They  lost  $  450.  What  was  each 
partner's  share  of  the  loss  ? 

11.  A,  B,  and  C  formed  a  partnership  for  one  year.  A 
put  in  $  850,  and  at  the  end  of  7  months  withdrew  $  300 ;  B 
put  in  $  1240,  and  at  the  end  of  5  months  withdrew  $  540 ; 
C  put  in  $  950,  and  at  the  end  of  9  months  added  $  200. 
They  lost  $  530.    What  was  each  partner's  share  of  the  loss  ? 

12.  A,  B,  and  C  entered  into  partnership.  A  put  in  i  of 
the  capital  for  ^  of  the  year ;  B  ^  of  the  capital  for  |  of 
the  year ;  and  C  the  balance  for  f  of  the  year.  They  gained 
$  6675.     What  was  each  partner's  share  of  the  profits  ? 

13.  On  Jan.  1st,  Lowe  and  Martin  entered  into  partner- 
ship for  one  year ;  Lowe  putting  in  $  5500,  and  Martin  $  7000. 
On  May  1st  they  admitted  Neal  with  a  capital  of  $  3500. 
On  Sept.  1st,  Lowe  put  in  $  2000,  and  on  Nov.  1st,  Martin 
withdrew  $1000.  They  gained  $5060.  What  was  each 
partner's  share  of  the  profits  ? 


210  ARITHMETIC. 


XVIII.    PERCENTAGE. 

283.  The  expression  Per  Cent  signifies  hundredths. 
Thus,  7  per  cent  of  any  number  is  the  same  as  .07  of  the 

number. 

284.  The  symbol  %  stands  for  per  cent. 

Thus,  3%  means  3  per  cent;  11^%  means  17J  per  cent; 
and  so  on. 

285.  The  Base  is  the  number  of  which  the  per  cent  is 
taken. 

The  Rate  Per  Cent  is  the  fraction  which  denotes  how  many 
hundredths  are  taken. 

The  Percentage  is  the  result. 

For  example,  5%  of  f  25  =  .05  of  $  25  =  $  1.25. 

In  this  case,  the  base  is  ^25,  the  rate  per  cent  is  5%  or 
.05,  and  the  percentage  is  $  1.25. 

286.  Percentage  is  the  process  of  computing  by  per  cents. 

EXAMPLES. 

287.  1.  Express  f  %  as  a  decimal.     Result.  .OOf,  or  .0075. 

2.  Express  .08%  as  a  decimal.  Result.  .0008. 

Express  each  of  the  following  as  a  decimal : 

3.  9%.  6.871%.  9.    1%.  12.1.6%. 

4.  58%.  7.  11J%.  10.  1%%.  13.  .07%. 

5.  136%.  8.     5f%.  11.  i^%.  14.  .069%. 

288.  To  express  a  Rate  Per  Cent  as  a  Common  Fraction. 
Example.     Express  42^%  as  a  common  fraction. 

.ofi^      424     ^^     300      3      . 


PERCENTAGE. 


211 


The  following  table  of  per  cents  with  their  fractional 
equivalents  should  be  carefully  committed  to  memory ;  the 
results  are  left  as  exercises  for  the  student. 


Per 

Frac- 

Per 

Frac- 

Per 

Frac- 

Per 

Frac- 

cent. 

tion. 

cent. 

tion, 

cent. 

tion. 

cent. 

tion. 

61- 

tV 

20 

i 

40 

f 

70 

tV 

H 

tV 

25 

i 

50 

i 

75 

1 

10 

tV 

30 

A 

60 

f 

80 

i 

12| 

i 

33^ 

\ 

Q2\ 

1 

87| 

i 

16f 

i 

371 

f 

66| 

f 

90 

T% 

EXAMPLES. 

Express  each  of  the  following  as  a  common  fraction  in  its 
lowest  terms : 


1.  48%.     4.  43f%.     7.  88|%.     10.  1^-%.     13.  7.5%. 

2.  175%.     5.  70f%.     8.  1%.         11.  11%.     14.  .96%. 

3.  6|%.     6.  47H%.  9-  f%-         12.  ¥%•     15.  .625%. 
289.  To  express  a  Common  Fraction  as  a  Rate  Per  Cent. 
1.   Express  f  as  a  rate  per  cent. 


8)3.00 


Dividing  3  by  8  to  two  places  of  decimals,  we  obtain  the 


3J1      result  .37 1. 

^         Whence,  f  =  371  %,  Ans. 


EXAMPLES. 
Express  each  of  the  following  as  a  per  cent: 


2. 

f 

6. 

u- 

10. 

H- 

14. 

Th- 

18. 

m- 

3. 

i. 

7. 

A- 

11. 

U- 

15. 

sis- 

19. 

sir- 

4. 

f- 

8. 

A- 

12. 

if- 

16. 

"T2T- 

20. 

m- 

5. 

A- 

9. 

fi- 

13. 

xi^- 

17. 

860- 

21. 

m- 

212  ARITHMETIC. 

290.  To  find  the  Percentage  when  the  Base  and  Rate  are 
given. 

1.  What  is  38%  of  f  15? 

$15 
.38 

-j  20  38  %  is  the  same  as  .38. 

^  ^  Multiplying  $  15  by  .38,  the  result  is  $5.70. 

$5.70,  Ans. 

2.  What  is  77^%  of  288  ? 

100      100     9* 

32 
Whence,  ^of  ?^^  =  224,  Ans. 

V 

Note  1.  It  is  not  easy  to  indicate  by  a  rule  when  it  is  better  to 
use  the  method  of  Ex.  1,  and  when  that  of  Ex.  2, 

But  in  general  we  may  say  that  if  the  rate  can  be  readily  expressed 
as  a  decimal,  the  method  of  Ex.  1  is  preferable,  unless  the  base  is  a 
common  fraction  or  mixed  number  which  cannot  be  readily  expressed 
as  a  decimal. 

In  all  other  cases,  the  method  of  Ex.  2  should  be  used. 

The  per  cent  should  always  be  expressed  mentally  as  a  common 
fraction  whenever  possible.     (See  Table,  Art.  288.) 

Note  2.  To  find  any  per  cent  of  a  compound  number  (Art.  159), 
it  is  usually  better  to  express  the  base  in  terms  of  the  lowest  given 
denomination  (Art.  157). 

EXAMPLES. 
What  is 

3.  7%  of  61.3?  9.  .8%  of  $283.75? 

4.  148%  of  $5.25?  10.  |%of6i|? 

5.  6i%  of  992  bu.  ?  11.  581%  of  £171  12s.  ? 

6.  871%  of  If  ?  12.  1361%  of  f  J  ? 

7.  791%  of  1^?  13.  .55%ofl4i|-? 

8.  71f  %  of  If  ?  14.  4%  of  283  ft.  4  in.  ? 


PERCENTAGE.  213 

15.  A  tradesman  bought  goods  for  ^350,  and  sold  them 
at  a  profit  of  18%.     How  much  did  he  receive  for  them  ? 

16.  A  dealer  purchased  goods  to  the  amount  of  $  1245, 
and  sold  them  at  a  loss  of  33^%.  How  much  did  he  lose 
by  the  operation  ? 

17.  A  bankrupt  settled  with  his  creditors  by  paying  to 
each  of  them  55%  of  what  he  owed  him.  How  much  was 
received  by  a  creditor  to  whom  he  owed  $  1732.80  ? 

18.  A  tradesman  bought  goods  for  $586.20,  and  sold 
them  at  a  loss  of  15%.  How  much  did  he  receive  for 
them  ? 

19.  A  man  at  his  death  bequeathed  66  J  %  of  his  property 
of  $  13587  to  his  wife ;  the  balance  to  be  divided  equally 
between  his  four  children.  How  much  did  each  child 
receive  ? 

20.  A  man  sold  a  horse  and  carriage  for  $  648.72,  receiv- 
ing 41|%  of  the  amount  in  cash.  How  much  was  still  due 
him? 

21.  A  merchant  bought  goods  for  $9675.30,  and  sold 
them  at  a  profit  of  23 J%.  How  much  did  he  gain  by  the 
operation  ? 

22.  If  52^%  of  the  population  of  a  certain  city  of  29844 
inhabitants  are  females,  what  is  the  number  of  males  in  the 
city? 

23.  My  income  is  $2160.  I  pay  43J%  of  it  for  living 
expenses,  15%  for  life-insurance,  and  invest  the  balance  in 
stocks.     What  sum  do  I  invest  in  stocks  ? 

24.  A  dealer  bought  125  barrels  of  flour  at  $6.72  each. 
After  reserving  29  for  his  own  use,  he  sold  the  rest  at  such 
a  price  that  he  gained  8  %  on  his  original  outlay.  At  what 
price  per  barrel  did  he  sell  the  flour  ? 

25.  T  bought  an  acre  of  land  for  $  1089,  and  sold  it  at  a 
profit  of  20%  J     For  what  price  per  foot  did  I  sell  it  ? 


214  ARITHMETIC. 

26.  If  52%  of  a  certain  ore  is  lead,  and  1J%  of  the 
balance  is  silver,  how  many  pounds  of  each  metal  are  there 
in  a  ton  of  the  ore  ? 

27.  A  man  who  has  $2112  in  the  bank,  draws  out  8J% 
of  this  sum,  and  afterwards  draws  out  6}%  of  the  balance. 
How  much  money  has  he  still  in  the  bank  ? 

28.  A  bought  goods  to  the  amount  of  $  315.20.  He  sold 
them  to  B  at  an  advance  of  ^^fjoi  and  B  then  sold  them  to 
C  at  12  L%  less  than  his  outlay.  How  much  did  C  pay  for 
them? 

29.  The  population  of  a  certain  town  in  1860  was  3552. 
It  increased  91|%  from  1860  to  1880,  and  decreased  37i% 
from  1880  to  1890.  How  much  greater  was  its  population 
in  1890  than  in  1860  ? 

30.  A  certain  metal,  when  heated,  expands  -^^%  for  each 
degree  Fahrenheit.  How  much  will  a  bar  of  the  above 
metal,  30  feet  long,  increase  in  length,  if  its  temperature  be 
raised  80  degrees  ? 

31.  A  grocer  mixes  ^^  pounds  of  coffee  worth  37  cents  a 
pound,  with  49  pounds  worth  43  cents  a  pound.  At  what 
price  per  pound  must  he  sell  the  mixture  so  as  to  gain 
16|%  ? 

32.  A  gentleman  has  the  sum  of  $6480  to  invest;  he 
puts  26|%  of  it  into  railway  bonds,  38f  %  of  what  remains 
into  stocks,  and  45|%  of  the  balance  into  real  estate.  How 
much  has  he  left  ? 

291.  To  find  the  Base  when  the  Percentage  and  Rate  are 
given. 

Since  the  Percentage  is  equal  to  the  Base  multiplied  by 
the  Kate,  it  follows  that  the  Base  may  be  found  by  dividing 
the  Percentage  by  the  Mate. 


PERCENTAGE.  215 

1.  ^  11.64  is  24%  of  what  sum  ? 

.24)$  11.64 ($  48.50,  ^rts. 

^  ^  24  %  is  the  same  as  .24. 

2  04  Dividing  $11.64  by  .24,  the  result 

192  is  $48.50. 

120 

2.  85  is  831%  of  what  number  ? 

83^^250^250^5 
100      100      300     6' 

17 
Whence,  §^  =  ^^  X  ^  =  102,  ^ws. 

f  P 

Note  1.  If  the  rate  can  be  readily  expressed  as  a  decimal,  the 
method  of  Ex.  1  is  preferable,  unless  the  percentage  is  a  common 
fraction  or  mixed  number  which  cannot  be  readily  expressed  as  a 
decimal. 

In  all  other  cases,  the  method  of  Ex.  2  should  be  used,- 

The  per  cent  should  always  be  expressed  mentally  as  a  common 
fraction  whenever  possible. 

Note  2.  If  the  percentage  is  a  compound  number  (Art,  159),  it  is 
usually  better  to  express  it  in  terms  of  the  lowest  given  denomination. 


EXAMPLES. 

Find  the  number  of  which 

3.  2.916  is  9%.  9.  6^is|%. 

4.  $9.99  is  54%.  10.  I3^is47|%. 

5.  J  is  371%.  11.  ^3^  is  9.5%. 

6.  8°  17'  is  46|%.  12.  6  lb.  11  oz.  av.  is  ^%. 

7.  I^is41|%.  13.  4^isl45f%. 

8.  35  ft.  9  in.  is  75%.  14.  $  1.24  is 


15.    A  merchant  sold  goods  for  f  416.52,  which  was  17% 
more  than  they  cost  him.     How  much  did  they  cost  him  ? 


216  ARITHMETIC. 

If  he  sold  the  goods  for  .17  more  than  they  cost  him,  he  must  have 
sold  them  for  1.17  of  what  they  cost  him. 

Then,  since  $416.52  is  1.17  of  what  the  goods  cost,  we  divide 
$416.62  by  1.17  to  obtain  the  result. 

1.17)$ 4.16.52($ 356,  Ans. 
351 


655 

585 


702 

16.  The  population  of  a  certain  town  decreased  12^% 
from  1880  to  1890.  If  the  number  of  inhabitants  in  1890 
was  2877,  what  was  the  number  in  1880  ? 

If  the  population  decreased  12|%,  or  i,  it  must  be  |  as  great  in  1890 
as  in  1880. 

Then,  since  2877  is  |  of  the  population  in  1880,  we  divide  2877  by  | 
to  obtain  the  result. 

2877       ^^^       " 


^  =  ?^yTx  ^  =  3288,  ^715. 


17.  The  sum  of  two  numbers  is  96 ;  and  one  of  them  is 
66^  %  greater  than  the  other.     What  are  the  numbers  ? 

The  greater  number  is  66|  %,  or  |,  greater  than  the  less  ;  it  is  there- 
fore f  of  the  less  number. 

Then  thesum  of  the  two  numbers  must  be  |  +  f ,  or  |  of  the  less 
number. 

?^  _  Qg  X  5  =  36  Dividing  96  by  f ,  we  find  the  less  number  to 

1^  ^  *     be  36  ;  and  the  greater  number  is  96  minus  36, 

96-36  =  60.    ^^^^^^^'' 

18.  A  man  having  lost  14  %  oi  his  money,  finds  that  he 
has  $  71.81  left.     How  much  had  he  at  first  ? 

19.  I  spend  $  1162  a  year,  which  sum  is  87J  %  of  my 
income.    What  is  my  income  ? 


PERCENTAGE.  217 

20.  What  number  increased  by  58|^%  of  itself  is  equal 
to  608  ? 

21.  What  number  diminished  by  36%  of  itself  is  equal 
to  420  ? 

22.  A  tradesman  gained  16%  by  selling  goods  for  $493. 
How  much  did  the  goods  cost  him  ? 

23.  What  sum  of  money  increased  by  24%  of  itself  is 
equal  to  $  77.50  ? 

24.  A  merchant  lost  8^%  by  selling  goods  for  $952. 
How  much  did  the  goods  cost  him  ? 

25.  What  number  diminished  by  71f  %  of  itself  is  equal 
to  A? 

26.  A  dealer  lost  12%  by  selling  goods  for  $260.70. 
How  much  money  would  he  have  gained  by  selling  them 
for  $  342  ? 

27.  Cloth,  when  sponged,  shrinks  1|%  of  its  length.  If 
a  certain  piece  of  cloth  has  lost  2  ft.  9  in.  by  sponging,  what 
was  its  original  length  ? 

28.  The  number  of  voters  in  a  certain  city  is  7925,  which 
is  4i%  more  than  the  number  two  years  ago.  How  many 
were  there  two  years  ago  ? 

29.  A  merchant  gained  6|%  by  selling  goods  for 
$  1372.16.     How  much  did  the  goods  cost  him  ? 

30.  At  a  charity  concert  which  realized  $  200.20  for  the 
poor,  the  expenses  were  35%  of  the  receipts.  What  were 
the  expenses  ? 

31.  A  brigade  of  soldiers  lost  33i%  of  its  numbers  in 
one  battle,  and  6J%  of  the  remainder  in  another,  and  had 
1830  men  left.     How  many  were  there  at  first  ?.. 

32.  There  are  297  pupils  in  a  certain  school,  and  the 
number  of  boys  is  20%  greater  than  the  number  of  girls. 
How  many  are  there  of  each  sex  ? 


218  ARITHMETIC. 

33.  A  merchant  who  owned  40%  of  a  vessel,  sold  64% 
of  his  share  for  $  4480.  What  was  the  value  of  the  entire 
vessel  at  the  same  rate  ? 

34.  The  difference  between  two  numbers  is  425,  and  one 
of  them  is  b2^^^o  greater  than  the  other.  What  are  the 
numbers  ? 

35.  A  merchant  sells  goods  at  a  profit  of  21%,  and  clears 
$  385.35.     What  was  the  selling  price  of  the  goods  ? 

36.  A  gentleman  invested  58^%  of  a  certain  sum  in  city- 
bonds,  and  53^%  of  the  balance  in  mortgages,  and  had 
^  1106  left.     How  much  had  he  at  first  ? 

37.  A  tradesman  sold  goods  for  $655.50,  losing  20|% 
of  what  they  cost  him.  At  what  price  should  the  goods 
have  been  sold  so  as  to  gain  20|%  ? 

38.  A  merchant  loses  $  461.50  by  selling  a  lot  of  goods 
at  31 J  %  less  than  cost.  What  was  the  selling  price  of  the 
goods  ? 

39.  The  population  of  a  city  increased  25%  from  1870 
to  1880,  and  28%  from  1880  to  1890.  If  the  population  in 
1890  was  92184,  what  was  the  population  in  1870  ? 

40.  A  speculator  bought  a  number  of  shares  of  railway 
stock.  The  price  of  the  stock  first  advanced  24%,  and 
then  fell  off  16%,  and  he  sold  for  $  1302.  What  was  the 
original  cost  of  the  shares  ? 

292.  To  find  what  Per  Cent  one  Number  is  of  Another. 

1.    What  per  cent  of  $  32  is  f  18  ? 
$32)$18.00(.56A  =  56i%,  A,^s.      ^.^^.^^^^  ^^^  ^^  ^^^^  ^^^ 

quotient  is  .56|^. 
^^  Then,  $18  is  .56^  of  $32, 

L^  or56p/oOf  $32. 

o 

Note  1.  The  division  should  always  be  carried  out  to  two  places 
of  decimals. 


PERCENTAGE.  219 

2.  What  per  cent  of  56  is  35  ? 
Dividing  35  by  56,  we  have 

ff  =  |=62i%,  ^ns. 

3.  What  per  cent  of  75  is  f  ? 

75     5     J^      125     500      100      ^^^^^ 
25 

In  the  above  example,  we  multiply  both  terms  of  the 
fraction  -^--^  by  4,  in  order  to  make  its  denominator  a  mul- 
tiple of  100. 

Note  2.  If  the  method  of  Ex.  2  is  employed,  the  fraction  should 
always  be  expressed  mentally  as  a  per  cent  whenever  possible. 

Note  3.  If  both  the  given  numbers  are  compound^  it  is  usually 
better  to  express  them  in  terms  of  the  lowest  given  denomination. 

EXAMPLES. 

What  per  cent  of 

4.  105  is  7  ?  10.  $  3.25  is  $  3.51  ? 

5.  120  is  48?  11.  94pk.4qt.is60pk.3qt.? 

6.  ^72  is  $39?  12.  13iis4i? 

7.  911  is  674.14?  13.  105  is  J? 

8.  72  is  66  ?  14.  8^  ft.  is  42  in.  ? 

9.  £  3  8s.  is  £  2  lis.  ?  15.  83^  is  ^  ? 

16.  If  a  merchant  buys  goods  for  $  420,  and  sells  them 
for  $  595,  what  does  he  gain  per  cent  ? 

17.  If  a  tradesman  buys  goods  for  $  352,  and  sells  them 
for  $  308,  what  does  he  lose  per  cent  ? 

18.  An  article  is  composed  of  19  parts  silver,  and  6  parts 
copper.  What  per  cent  of  the  article  is  silver?  What 
per  cent  is  copper?  What  per  cent  is  the  copper  of  the 
silver  ? 


220  ARITHMETIC. 

19.  The  number  of  voters  in  a  certain  city  increased 
from  4644  to  5547.     What  was  the  gain  per  cent  ? 

20.  Air  is  a  mixture  of  nitrogen  and  oxygen.  If  the 
nitrogen  is  79%  of  the  mixture,  what  per  cent  is  the  oxygen 
of  the  nitrogen  ? 

21.  If  my  income  is  $2745,  and  my  expenses  $2135, 
what  per  cent  of  my  salary  do  I  save  ? 

22.  From  a  barrel  containing  24  gallons  of  water,  18 
quarts  leaked  out.     What  per  cent  of  the  water  was  lost  ? 

23.  If  2048  of  the  16384  inhabitants  of  a  city  are  foreign- 
born,  what  per  cent  of  the  population  is  native  ? 

24.  If  a  merchant  buys  goods  for  five-sixths  of  what  he 
sells  them  for,  what  per  cent  does  he  gain  ? 

25.  If  from  45  tons  of  ore  there  are  obtained  675  pounds 
of  silver,  what  per  cent  of  the  ore  is  silver  ? 

26.  A  miller  mixes  8  barrels  of  flour,  worth  $6.25  a 
barrel,  with  12  barrels,  worth  $6.50  a  barrel,  and  sells  the 
whole  for  $ 6.56  a  barrel.     What  per  cent  does  he  gain? 

27.  A  man  having  $  203.40  in  the  bank,  withdrew  $  33.90. 
What  per  cent  of  the  original  sum  remained  ? 

28.  What  per  cent  is  gained  by  buying  apples  at  $  1.25  a 
bushel,  and  selling  them  at  5  cents  a  quart  ? 

29.  A  grocer  mixes  12  pounds  of  tea,  worth  70  cents  a 
pound,  with  15  pounds,  worth  84  cents  a  pound,  and  sells 
the  mixture  at  91  cents  a  pound.  What  per  cent  does  he 
gain? 

30.  If  one-seventh  of  the  price  received  for  an  article  is 
gain,  what  is  the  gain  per  cent  ? 

31.  I  sold  a  house  for  $5376,  which  was  28%  more  than 
I  gave  for  it.  If  I  had  sold  it  for  $  4998,  what  per  cent 
should  I  have  gained  ? 


PERCENTAGE.  221 

32.  I  purchased  an  acre  of  land  for  $  2100,  and  sold  it 
at  5^  cents  a  square  foot.     What  per  cent  did  I  gain  ? 

33.  A  grocer  uses  a  false  weight  of  15  ounces  instead  of 
a  pound.  What  per  cent  does  he  gain  by  his  dishonesty  ? 
What  per  cent  do  his  customers  lose  ? 

34.  If  -fj  of  the  price  received  for  an  article  is  gain, 
what  is  the  gain  per  cent  ? 

36.  A  merchant  bought  a  lot  of  goods.  The  price  at  first 
declined  12%,  and  then  advanced  16%,  and  he  then  sold 
out.     What  per  cent  did  he  gain  ? 

36.  A  grocer  bought  180  barrels  of  flour  at  $  6  a  barrel, 
and  sold  60  barrels  at  a  loss  of  8J%.  At  what  per  cent 
above  cost  must  he  sell  the  remainder,  so  as  to  gain  10% 
on  the  entire  transaction  ? 

37.  If  291%  is  lost  by  selling  goods  for  $275.40,  what 
per  cent  would  be  gained  by  selling  them  for  $534.60? 

APPLICATIONS  OF    PERCENTAGE. 

293.  Trade  Discount. 

Trade  Discount  is  a  reduction  made  in  the  price  of  an 
article.  It  is  usually  a  certain  per  cent  of  the  list  price  of 
the  article ;  but  in  some  cases,  goods  are  sold  subject  to 
several  discounts. 

The  Net  Price  of  an  article  is  the  list  price,  minus  the 
discount. 

EXAMPLES. 

294.  1.  Findthenet  amount  of  a  bill  for  $563.20, 12^% 
off  for  cash. 

8)  $563.20  12^  0/^  of  Z  563.20  is  i  of  $  563.20,  or  $  70.40. 

70.40  Subtracting   $70.40   from   ^563.20,   the   net 

$  492.80,  Ans.     amount  is  1 492.80. 


222  ARITHMETIC. 

2.  Find  the  net  amount  of  a  bill  for  $460,  subject  to  a 
discount  of  15^,  and  5%  off  for  cash. 

$460 

1^  Making  a  discount  of  15  %  is  the  same  thing 

23  00  as  taking  85%  of  the  face  of  the  bill,  which  is 

368  0  $391.00. 
20)  $391.00  5  %  of  $391  is  ^V  of  $391,  or  1 19.55  ;  which, 

19.55  subtracted  from  $  391,  leaves  $  371.45. 

$371.45,  Ans. 

3.  A  man  bought  goods  to  the  amount  of  $54.90,  and 
obtained  a  discount  of  16|%  for  cash.  How  much  did  he 
pay  for  the  goods  ? 

4.  Find  the  net  amount  of  a  bill  for  $231,  subject  to  a 
discount  of  8^%,  and  4%  off  for  cash. 

5.  A  tradesman  marks  a  certain  article  $  19.60 ;  but  he 
makes  a  discount  of  20%  from  this  price,  and  then  a  dis- 
count of  6J%  for  cash.  What  price  does  he  receive  for  the 
article  ? 

6.  A  dealer  sells,  at  a  discount  of  8%  from  his  list  price, 
a  certain  article  marked  $  6.00.  If  he  still  makes  a  profit 
of  38%,  what  was  the  cost  of  the  article  ? 

7.  How  much  must  a  merchant  mark  an  article  costing 
$6.50,  so  as  to  be  able  to  sell  it  at  9%  below  the  list  price, 
and  still  make  a  profit  of  12%  ? 

8.  How  much  must  a  dealer  mark  an  article  costing 
$3.50,  so  as  to  be  able  to  sell  it  at  a  discount  of  6|%  from 
the  list  price,  and  still  make  a  profit  of  16%  ? 

9.  Find  the  net  amount  of  a  bill  for  $  25.60,  subject  to 
discounts  of  25%,  15%,  and  12^%. 

10.  A  tradesman  marks  goods  25%  above  cost,  and  then 
sells  them  at  a  discount  of  23%  from  the  list  price.  Does 
he  gain  or  lose,  and  what  per  cent  ? 


PERCENTAGE.  223 

11.  How  much  above  cost  must  a  merchant  mark  an 
article  costing  him  f  3.24,  so  as  to  be  able  to  sell  it  at  a 
discount  of  10%  from  the  list  price,  and  still  make  a  profit 
of  25%  ? 

12.  Find  the  net  amount  of  a  bill  for  $  5265,  subject  to 
discounts  of  33i%,  16%,  and  10%. 

13.  A  dealer  bought  silk  at  $  1.40  a  yard.  What  price 
per  yard  must  it  be  marked  in  order  that  he.  may  be  able  to 
make  a  discount  of  20%  from  the  list  price,  and  still  make 
a  profit  of  20%? 

14.  A  tradesman  marks  an  article  25%  above  cost,  and 
then  sells  it  at  a  discount  of  12|-%  from  the  list  price. 
What  per  cent  does  he  gain  ? 

15.  How  much  above  cost  must  a  merchant  mark  an 
article  costing  S  20.79,  so  as  to  be  able  to  sell  it  at  a  discount 
of  6|%  from  the  list  price,  and  still  make  a  profit  of  11^%  ? 

16.  Find  the  net  amount  of  a  bill  for  f  19.20,  subject  to 
discounts  of  16|%,  7^%,  6J%,  and  4%. 

17.  What  per  cent  above  cost  must  a  merchant  mark  an 
article,  in  order  to  be  able  to  sell  it  at  a  discount  of  16% 
from  the  list  price,  and  still  make  a  profit  of  19%  ? 

If  the  article  is  sold  at  a  discount  of  16  %  from  the  list  price,  it  must 
be  sold  for  84  %  of  the  list  price. 

If  it  sold  for  19%  above  cost,  it  must  be  sold  for  119%  of  the  cost 
price. 

Then,  since  ^Vo  ^f  ^^^  ^^^^  price  is  {^^  of  the  cost  price,  the  list  price 
must  be  Y^  of  |^§,  or  -^^r  of  the  cost  price. 

W  =  H  =  i4if%. 

That  is,  the  list  price  is  141f  %  of  the  cost  price. 

Then  the  article  must  be  marked  41|  %  above  cost,  Ans. 

18.  At  what  per  cent  above  cost  must  a  tradesman  mark 
an  article,  in  order  to  be  able  to  sell  it  for  7%  less  than  the 
list  price,  and  still  make  a  profit  of  24%  ? 


224  ARITHMETIC. 

19.  What  per  cent  above  cost  must  a  merchant  mark 
goods,  so  as  to  be  able  to  make  a  discount  of  18%  from  the 
list  price,  and  still  realize  a  profit  of  23%  ? 

20.  What  per  cent  above  cost  must  goods  be  marked,  to 
allow  a  discount  of  10%  from  the  list  price,  and  still  make 
a  profit  of  15%  ? 

21.  What  per  cent  above  cost  must  goods  be  marked,  to 
allow  a  discount  of  20%,  and  still  make  a  profit  of  9%  ? 

22.  What  per  cent  above  cost  must  goods  be  marked,  to 
allow  a  discount  of  12%,  and  still  make  a  profit  of  21%  ? 

295.  Commission  and  Brokerage. 

Commission,  or  Brokerage,  is  the  compensation  received 
by  an  agent  for  transacting  business. 

It  is  usually  a  certain  per  cent  of  the  amount  received, 
when  selling,  and  of  the  amount  paid,  when  buying  goods. 

The  agent  is  variously  styled  2i  factor,  broker,  collector,  or 
commission  merchant. 

The  amount  left  over  after  paying  the  commission  and 
other  charges  is  called  the  net  proceeds, 

EXAMPLES.  ' 

296.  1.  An  agent  has  sold  for  me  goods  to  the  amount 
of  $2300.  His  commission  is  2-J%,  and  the  charges  for 
carting  and  storage  are  $  16.25.     How  much  is  due  me  ? 

$2300 
.025 


11500 
46  00 


2|%  of  $2300  is  f  57.50,  which  is  the  amount 
of  the  commission. 
$57.50  Adding  to  this  the  charges,  $16.25,  the  total 

^^•-^^  amount  to  be  deducted  is  $  73.75. 

$  73.75  Subtracting  this  from  $  2300,  tlie  net  proceeds 

$2300.  are  $2226.25. 

73.75 
$  2226.25,  ^ns. 


PERCENTAGE.  226 

2.  An  agent  receives  $  1500  to  invest,  after  deducting 
a  commission  of  3%.  What  sum  can  he  invest,  and  what 
is  the  amount  of  his  commission  ? 

1.03)  f  1500(^1456.31 
103 

470  Since  the  commission  is  3  %  of  the  sum 

412  invested,  the  amount  received  is  103  %  of 

goA  the  sum  invested. 

5j^5  Then,  if  $il500  is  1.03  times  the  sum 

~n^  invested,  we  divide  $1500  by  1.03,  giving 

/^-jo  f  1456.31  +  ,  which  is  tlie  sum  invested; 

and  $  1500  -  $  1456.31  =  .f  43.69,  the  com- 
mission, Ans. 


320 
309 
110 


3.  An  agent  sells  $576  worth  of  cloth.  What  is  his 
commission  at  3J%  ? 

4.  A  commission  merchant  sells  for  me  goods  to  the 
amount  of  $5146.50.  His  commission  is  2%,  and  there 
are  charges  of  $  78.75  for  storage.     How  much  is  due  me  ? 

5.  A  factor  sells  a  consignment  of  goods  for  $  254,  and, 
after  taking  out  his  commission,  remits  $  241.30  to  the  con- 
signor.    What  is  his  rate  of  commission  ? 

6.  A  broker  bought  for  me  $8500  worth  of  securities, 
at  a  brokerage  of  i%.     What  was  the  brokerage? 

7.  A  commission  merchant  receives  $551.20  to  invest, 
after  deducting  a  commission  of  6%.  What  sum  can  he 
invest,  and  what  is  the  amount  of  his  commission  ? 

8.  A  broker  received  $1000  to  invest,  after  deducting 
a  commission  of  1%.  What  sum  can  he  invest,  and  what 
is  the  amount  of  his  commission  ? 

9.  An  agent  sold  goods  to  the  amount  of  $  9220,  receiv- 
ing $  404.25,  which  included  his  commission,  and  a  charge 
for  freight  and  storage  of  $58.50.  What  was  his  rate  of 
commission  ? 


226  ARITHMETIC. 

10.  A  commission  merchant  receives  $356.15  to  invest 
in  cloth  at  85  cents  a  yard,  after  deducting  his  commission 
of  4|%.     How  many  yards  can  he  buy? 

11.  What  amount  must  a  factor  receive  in  order  that  he 
may  be  able  to  invest  $2400,  after  deducting  his  commis- 
sion of  4%  ? 

12.  An  agent  charged  4^%  for  buying  goods.  If  his 
commission  was  $  333.20,  what  did  he  pay  for  the  goods  ? 

13.  A  factor  received  a  sum  of  money  to  invest,  after 
deducting  his  commission  of  5%.  He  invested  $2251.60. 
What  sum  did  he  receive  ? 

14.  What  sum  must  I  send  to  a  broker  in  order  that  he 
may  be  able  to  invest  $  436,  after  deducting  his  brokerage 
of  i%  ? 

15.  An  agent  received"  $  2500  to  invest,  after  deducting 
his  commission  of  2J%.  What  sum  can  he  invest,  and 
what  is  the  amount  of  his  commission  ? 

16.  A  broker,  who  received  f  %  brokerage  for  selling 
some  securities,  paid  $8337  to  the  owner  as  net  proceeds. 
For  what  sum  were  the  securities  sold  ? 

17.  What  are  the  net  proceeds  on  the  sale  of  368  barrels 
of  flour  at  $6.50  a  barrel,  the  commission  being  S^%,  and 
the  charges  for  freight  and  storage  41  cents  a  barrel  ? 

18.  A  merchant  received  $2194.50  to  invest  in  flour  at 
$  6.25  a  barrel,  after  deducting  a  commission  of  4J%.  How 
many  barrels  can  he  buy  ? 

19.  An  agent  sells  $  648  worth  of  goods,  receiving  a  com- 
mission of  1^%.  He  invests  the  net  proceeds  in  other 
goods,  charging  the  same  commission.  What  is  the  value 
of  the  goods  bought  ? 

20.  My  agent  has  sold  for  me  4000  yards  of  silk,  at 
$  1.69  a  yard.  His  charges  are  as  follows  :  —  commission, 
2^%  ;  guarantee,  1^%  ;  freight,  $  116.50.  How  much  is 
due  me  ? 


PERCENTAGE.  227 

21.  An  agent  sells  206  yards  of  cloth,  at  ^  2.50  a  yard, 
charging  2%  commission.  He  invests  the  net  proceeds  in 
woollens  at  $1.40  a  yard,  charging  3%  commission.  How 
many  yards  can  he  buy  ? 

22.  A  broker  received  $1113.84  to  invest  in  bonds  at 
$  85  each,  after  deducting  his  brokerage  of  |%.  How  many 
bonds  can  he  buy,  and  what  will  be  the  amount  of  his 
brokerage  ? 

23.  An  agent  sells  80  tons  of  coal  at  $  8.30  a  ton,  receiv- 
ing a  commission  of  If  %.  He  invests  the  net  proceeds  in 
flour,  at  $6.55  a  barrel,  charging  3f%  commission.  How 
many  barrels  can  he  buy  ? 

297.  Insurance. 

Insurance  is  security  against  loss. 

A  Policy  of  Insurance  is  a  written  contract  containing  an 
agreement  on  the  part  of  the  Insurer,  for  a  specified  con- 
sideration, to  pay  a  specified  sum  to  the  insured,  in  case  of 
loss  by  fire,  death,  shipwreck,  or  otherwise. 

The  Premium  is  the  sum  paid  for  insurance,  and  is 
usually  reckoned  as  a  certain  per  cent  of  the  amount 
insured. 

EXAMPLES. 

298.  1.  What  will  be  the  annual  premium  for  insuring 
$3500  on  my  life,  at  2^%  a  year  ? 

$3500 
.02^ 
70  00 

8  75 


$78.75,  Ans. 

2.  For  what  sum  must  goods  valued  at  $  5432  be  insured 
at  3%,  in  order  that,  in  case  of  loss,  the  owner  may  receive 
both  the  value  of  the  goods  and  the  premium  paid  for  the 
insurance  ? 


228  ARITHMETIC. 

Since  the  premium  is  3  %  of  the  sum  insured,  the  value  of  the  goods 
must  be  100  %  -  3  %,  or  97  %,  of  the  sum  insured. 

Then,  if  $  5432  is  .97  of  the  sum  insured,  the  sum  insured  is  1 5432 
divided  by  .97. 

.97)  f  5432  ($  5600,  ^ws. 
485 

582 

3.  What  is  the  premium  for  insuring  $1476  on  mer- 
chandise for  one  year  Sit  3\%  ? 

4.  What  is  the  premium  for  insuring  $  7550  on  a  vessel 
at  51%  ? 

5.  If  the  premium  for  insurance  at  1J%  is  $  23.25,  what 
is  the  amount  insured  ? 

6.  If  I  pay  a  premium  of  $  13.24  for  insuring  my  house 
for  $  3972,  what  is  the  rate  per  cent  of  the  insurance  ? 

7.  For  what  sum  must  a  cargo  worth  $  1634  be  insured 
at  5%,  in  order  that,  in  case  of  loss,  the  owner  may  receive 
both  the  value  of  the  cargo  and  the  premium  ? 

8.  A  barn  was  insured  for  $4200,  at  f%  per  annum. 
After  23  annual  premiums  had  been  paid,  the  barn  was 
burned.  What  did  the  insurance  company  lose,  if  no  allow- 
ance be  made  for  interest  ? 

9.  A  merchant  insured  a  cargo  worth  $  12560  for  ^  of 
its  value,  at  4%.  In  case  of  shipwreck,  what  would  be  his 
actual  loss  ? 

10.  I  paid  $  15.40,  including  $  1  for  the  policy,  for  insur- 
ing $  3600  on  a  house.     What  was  the  rate  of  insurance  ? 

11.  A  mill  was  insured  for  831%  of  its  value,  at  4|%, 
and  the  premium  was  $  228.  What  was  the  value  of  the 
mill? 

12.  For  what  sum  must  a  vessel  worth  $  9168  be  insured 
at  4^%,  in  order  that,  in  case  of  loss,  the  owner  may  receive 
both  the  value  of  the  vessel  and  the  premium  ? 


PERCENTAGE.  229 

13.  A  man  44  years  of  age  takes  out  a  life-policy  for 
$  15000,  for  the  benefit  of  his  wife,  at  the  yearly  rate  of 
2J%.  Should  his  death  occur  at  the  age  of  74,  how  much 
more  would  his  widow  receive  than  had  been  paid  in  annual 
premiums  ? 

14.  A  merchant  insures  some  goods  for  $  2750  at  3J%, 
the  amount  insured  covering  both  the  value  of  the  goods  and 
the  premium.     What  was  the  value  of  the  goods  ? 

15.  What  premium  must  be  paid  when  merchandise 
worth  ^  6091.50  is  insured  at  If  %,  so  as  to  cover  both  the 
value  of  the  goods  and  the  premium  ? 

16.  A  house  worth  $  8000  was  insured  for  -^  of  its  value 
by  three  companies.  The  first  took  i  of  the  risk  at  2i%, 
the  second  \  of  the  risk  at  2%,  and  the  third  the  remainder 
at  2J%.     What  was  the  total  premium  ? 

17.  A  shipowner  insures  a  cargo  for  ^8400,  at  5%,  the 
amount  insured  covering  both  the  value  of  the  cargo  and 
the  premium.     What  was  the  value  of  the  cargo  ? 

18.  If  merchandise  valued  at  ^6614.30  is  insured  for 
$  6872,  the  amount  insured  covering  both  the  value  of  the 
goods  and  the  premium,  what  is  the  rate  per  cent  of  the 
insurance  ? 

299.  Taxes. 

A  Tax  is  a  sum  of  money  assessed  upon  the  person,  real 
estate,  personal  property,  or  income  of  an  individual,  for 
public  uses. 

When  assessed  upon  the  person,  it  is  called  a  Poll  Tax, 
and  is  always  a  fixed  amount  for  each  male  citizen  of  legal 
age,  without  regard  to  his  property. 

Note.     A  Poll  Tax  is  not  assessed  in  every  state. 

When  assessed  upon  the  property  of  an  individual,  it  is 
called  a  Property  Tax,  and  is  usually  reckoned  as  a  certain 
rate  per  cent  of  his  taxable  property. 

This  rate  per  cent  is  called  the  Rate  of  Taxation. 


230  ARITHMETIC. 

It  is  expressed  either  as  a  rate  per  cent,  or  as  so  many 
dollars  on  $  1000. 

Thus,  li%  is  the  same  as  $  12.50  on  $  1000. 

300.  To  determine  the  rate  of  taxation,  the  whole  amount 
of  the  poll  taxes,  if  any,  is  deducted  from  the  entire  sum  to 
be  raised  by  taxation ;  the  remainder,  divided  by  the  value 
of  the  taxable  property,  is  the  rate  of  taxation. 

To  find  the  tax  to  which  any  individual  is  liable,  multiply 
his  taxable  property  by  the  rate  of  taxation,  and  add  to  the 
product  the  poll  tax,  if  any ;  the  result  will  be  the  entire 
tax  to  which  the  individual  is  liable. 

EXAMPLES. 

301.  1.  A  certain  town  whose  taxable  property  is 
$  1146000,  wishes  to  raise  $  14562  by  taxation.  There  are 
405  polls,  each  assessed  $  2.00.  What  will  be  the  rate  of 
taxation  ?  What  will  be  the  entire  tax  of  an  individual 
whose  property  is  valued  at  Jp  8500  ? 

Multiplying  $2.00  by  405,  we  have  $810,  the  whole  amount  of  the 
poll  taxes. 

Subtracting  this  from  $  14,562,  the  entire  sum  to  be  raised  by  taxa- 
tion, the  remainder  is  $13,752. 

Dividing  this  by  $1,146,000,  the  value  of  the  taxable  property,  the 
rate  of  taxation  is  .012,  or  $  12  on  $  1000. 

Multiplying  $8500  by  .012,  the  product  is  $102,  which  is  the 
individual's  property  tax. 

Adding  to  this  his  poll  tax,  $  2.00,  his  entire  tax  is  $  104. 

2.  If  a  town  whose  taxable  property  is  $  75400,  wishes 
to  raise  f  1017.90,  what  is  the  rate  of  taxation  ? 

3.  A  town  whose  taxable  property  is  $165000,  wishes 
to  raise  f  3419.50.  There  are  266  polls,  each  paying  $  2.00. 
What  is  the  rate  of  taxation  ? 

4.  A  man's  house  is  valued  at  $  3640,  and  he  pays  a  poll 
tax  of  $  1.50.  What  will  be  his  entire  tax,  at  the  rate  of 
$14.50  on  1 1000? 


PERCENTAGE.  231 

5.  What  tax  must  be  assessed  in  order  that  ^  4074  may 
be  left,  after  paying  a  commission  of  3%  for  collection? 

6.  What  tax  must  be  assessed  in  order  that  $  6895  may 
be  left,  after  paying  a  commission  of  l|-%  for  collection  ? 

7.  What  tax  will  be  paid  by  a  man  whose  house  is  val- 
ued at  ^5160,  and  personal  property  at  ^7815,  and  who 
pays  for  2  polls  at  $  1.50  each,  the  rate  of  taxation  being 
$16.40  on  $1000? 

8.  The  sum  of  $  3603.66  is  to  be  raised  in  a  town  whose 
taxable  property  is  $282640.  What  is  the  rate  of  taxa- 
tion ?  What  will  be  the  tax  on  a  piece  of  property  which 
is  valued  at  $8400? 

9.  What  tax  must  be  assessed  in  order  that  a  city  may 
receive  $  34718.25,  after  the  collector  deducts  his  commis- 
sion of  2f  %  ? 

10.  A  collector  turned  over  to  the  town  treasurer  the  sum 
of  $7968,  after  deducting  his  commission  of  4%.  What 
was  the  amount  of  his  commission  ? 

11.  A  town  whose  real  estate  is  valued  at  $  724600,  and 
personal  property  at  $  561400,  wishes  to  raise  $  15352.30. 
There  are  547  polls,  each  paying  $  1.50.  What  is  the  rate 
of  taxation  ? 

12.  At  what  rate  must  property  valued  at  $425000  be 
taxed,  in  order  that  the  sum  of  $6664  may  be  left,  after 
paying  a  commission  of  2%  for  collection  ? 

13.  The  taxes  assessed  in  a  certain  town  are  $  24000.  If 
If  %  commission  is  paid  for  all  taxes  actually  collected, 
and  only  95%  of  the  taxes  can  be  collected,  what  are  the 
net  proceeds  ? 

302.  Duties. 

Duties,  or  Customs,  are  taxes  levied  on  imported  goods. 
A  Specific  Duty  is  a  fixed  tax  upon  an  article  without 
regard  to  its  value. 


282  ARITHMETIC. 

An  Ad  Valorem  Duty  is  levied  at  a  certain  rate  per  cent 
of  the  cost  of  the  goods  in  the  country  from  which  they  are 
imported. 

Tare  is  an  allowance  made  for  the  weight  of  the  box, 
cask,  etc.,  containing  the  goods. 

Leajiage  is  an  allowance  for  the  loss  of  liquors  in  casks ; 
Breakage  is  an  allowance  for  the  loss  of  liquors  in  bottles. 

Oross  weight  is  the  entire  weight  of  the  goods  and  pack- 
ages, before  any  allowances  are  made;  Net  weight  is  the 
weight  after  all  allowances  have  been  made. 

EXAMPLES. 

303.  Note.  In  the  following  examples,  the  pound  sterling  is 
valued  at  $  4.8665,  and  the  franc  at  $0,193. 

1.  A  merchant  imported  550  yards  of  silk,  invoiced  at  7 
francs  a  yard.     What  is  the  duty  at  60%  ad  valorem  ? 

550  X  7  =  3850  francs,  the  value  of  the  invoice  in  francs. 
A  franc  being  valued  at  $  0, 193,  the  value  of  the  invoice  in  dollars  is 
3850  X  $0,193,  or  $743.05. 

Then  60  %  of  $  743.05  is  $445.83,  Ans. 

2.  What  is  the  duty,  at  2.2  cents  a  pound,  on  1100  pounds 
of  tin-plate,  tare  5%  ? 

Since  the  tare  is  5  %,  the  number  of  pounds  on  which  the  duty  is 
levied  is  100%  -  5  %,  or  95%,  of  1100  ;  that  is,  1045. 

At  $0,022  a  pound,  the  duty  on  1045  pounds  is  $22.99,  Ans. 

3.  What  is  the  duty,  at  8  cents  a  hundred  weight,  on  57 
casks  of  salt,  each  containing  175  pounds  ? 

4.  What  is  the  duty,  at  25%  ad  valorem,  on  20  watches, 
invoiced  at  104  francs  each  ? 

5.  What  is  the  duty,  at  45%  ad  valorem,  on  5  dozen  gold 
rings,  invoiced  at  £1  4s.  each  ? 

6.  What  is  the  duty,  at  35  cents  a  gallon,  on  150  dozen 
quart  bottles  of  olive  oil,  breakage  4%  ? 


PERCENTAGE.  233 

7.  An  importer  paid  $111.24,  including  an  ad  valorem 
duty  of  35%,  for  a  fur  cape.     What  was  the  invoice  price  ? 

8.  What  is  the  duty,  at  20  cents  a  gallon,  on  10  casks 
of  ale,  each  containing  31^  gallons,  leakage  3%  ? 

9.  What  is  the  duty,  at  60  cents  a  square  yard,  and  40% 
ad  valorem,  on  375  square  yards  of  rugs,  invoiced  at  48 
francs  a  square  yard  ? 

10.  What  is  the  duty,  at  $4.50  a  pound,  and  25%  ad 
valorem,  on  10000  cigars,  invoiced  at  $  6.00  a  hundred,  and 
weighing  12  pounds  to  the  thousand  ? 

11.  A  merchant  imported  a  lot  of  linen  clothing,  invoiced 
at  $1120,  and  paid  $577.50  duty,  tare  6J%.  What  was 
the  rate  of  duty  ? 

12.  What  is  the  duty,  at  60%  ad  valorem,  on  125  yards  of 
silk,  invoiced  at  5.52  francs  a  yard,  tare  3%  ? 

13.  What  is  the  duty,  at  $2.00  a  dozen,  and  30%  ad 
valorem,  on  600  table-knives,  invoiced  at  4s.  5d.  each  ? 

14.  What  is  the  duty,  at  44  cents  a  square  yard,  and  35% 
ad  valorem,  on  1000  yards  of  carpet,  f  of  a  yard  wide, 
invoiced  at  6s.  9d.  a  yard  ? 


234  ARITHMETIC. 


XIX.    INTEREST. 

304.  Interest  is  money  paid  for  the  use  of  money. 

The  Principal  is  the  money  for  the  use  of  which  interest 
is  paid ;  the  Amount  is  the  sum  of  the  principal  and  interest. 

305.  Interest  is  usually  reckoned  as  a  certain  rate  per 
cent  of  the  principal  for  one  year. 

This  rate  per  cent  is  called  the  Rate  of  Interest. 

Note.     It  will  be  understood  in  the  following  examples  that  the 
rate  is  for  one  year,  unless  some  other  time  is  specified. 

SIMPLE  INTEREST. 

306.  Simple  Interest  is  interest  on  the  principal  alone. 

307.  General  Method  for  Computing  Interest  at  any  Rate. 

In  computing  interest,  it  is  customary  to  regard  the  year 
as  consisting  of  12  months  of  30  days  each. 

EXAMPLES. 

1.   What  is  the  interest  of  ^  700  for  8  years  at  5^  ? 

$700 

.05  The  interest  for  one  year  is  .05  of  the  principal, 

$35^00  or  $35.00. 

Multiplying  this  by  8,  the  interest  for  8  years  is 
$280.00. 


8 


$280.00,  Ans. 

If  the  given  time  is  any  number  of  years  and  months, 
and  the  months  can  be  expressed  as  the  decimal  of  a  year, 
the  method  of  Ex.  1  may  be  used. 

If  the  months  cannot  be  expressed  as  the  decimal  of  a 
year,  it  is  better  to  reduce  the  given  time  to  months,  multiply 
the  interest  for  one  year  by  the  number  of  months,  and  divide 
the  result  by  12. 


INTEREST.  235 

2.   Find  the  interest  of  $255  for  5  y.  7  mo.  at  4^%. 

^^^^  $11,475  The  interest  for  one  year  is 

'^H  ^  .04|  times  $255,  or  $11,475. 

10  20                      80  325  5  y.  7  mo.  is  67  mo. 

1275  688  50  To   find    the    interest   for   67 


$11,475  12)768.825  mo.,  we  multiply  the  interest  for 

$64,068+  ^^®  y^^^  ^y  ^"'  ^^^  divide  the 

=  $  64.07,  Ans.      i"esult  by  12. 

Note  1.  In  dividing  by  12,  there  is  no  need  of  carrying  the  quotient 
beyond  the  third  place  of  decimals  ;  the  result  should  then  be  expressed 
to  the  nearest  cent,  the  number  of  cents  being  increased  by  1  if  the  fig- 
ure in  the  third  decimal  place  is  5  or  more  than  5.    (Compare  Art.  132. ) 

If  the  given  time  is  any  number  of  years,  months,  and 
days,  the  days  should  be  expressed  as  the  decimal  of  a 
month,  if  possible ;  otherwise  as  the  fraction  of  a  month. 

3.  Find  the  amount  of  $26.25  from  Dec.  20,  1886,  to 
June  2,  1890,  at  7%. 

$26.25  $1.8375 

07  13 

^1  oo7f;  -^^77^  By  the  method  of  Art.   165,  we 

—^         ^^'^^^  to  June  2,  1890,  is  3  y.  5  mo.  13  da., 

18375         3)2.38875  or41Hmo. 
73  500             ^  0.796+  We  then  multiply  the  interest  for 

^^^  one  year  by  41if,   and  divide  the 

12)76.1335  result  by  12. 

$  6.34= Interest.  Adding  the  principal  to  the  inter- 

26.25= Principal.  ^®*'  *^®  amount  is  $32.59. 
$32.59= Amount,  Ans, 

Note  2.  The  work  of  multiplying  the  interest  for  one  year  by  ^f 
appears  in  the  right-hand  column  ;  we  multiply  $  1.8375  by  13,  move 
the  decimal  point  of  the  product  one  place  to  the  left,  and  divide  the 
result  by  3. 

Find  the  interest  of  : 

4.  $725  for  6  y.  at  5%. 

5.  $  173.25  for  3  y.  3  mo.  at  6%. 


236  ARITHMETIC 

6.  $  476  for  1  y.  7  mo.  at  2\%. 

7.  $  93.41  for  8  mo.  21  d.  at  7%. 

8.  $  591.68  for  4  y.  10  mo.  18  d.  at  3f  %. 

9.  $  227.75  for  2  mo.  5  d.  at  6J%. 

10.  ^  81.34  for 5  y.  9  mo.  23  d.  at  41%. 

11.  $  159  from  Jan.  3,  1889,  to  Dec.  18, 1891,  at  3%. 

12.  $  918.37  from  Nov.  20, 1883,  to  June  13, 1885,  at  3i%. 

13.  $  78.96  from  July  17, 1886,  to  Nov.  27,  1890,  at  5i%. 

14.  $  34.70  from  Aug.  29,  1891,  to  Aug.  7,  1892,  at  8%. 

15.  $  209.50  from  Dec.  12,  1885,  to  May  23,  1889,  at  2%. 

16.  $  694.03  from  May  19,  1881,  to  July  6,  1888,  at  4%. 

Find  the  amount  of: 

17.  $  63.59  for  5  y.  9  mo.  at  3i%. 

18.  $  245.25  for  2  y.  5  mo.  at  5i%. 

19.  $  1869.84  for  6  y.  8  mo.  12  da.  at  2|%. 

20.  1 463  for  3  mo.  7  d.  at  4%. 

21.  $  3662.95  from  March  7, 1891,  to  Oct.  13, 1891,  at  6%. 

22.  $  851.32  from  April  16,  1883,  to  March  26,  1890,  at 
41%.       . 

23.  $  504.18  from  June  23, 1887,  to  Sept.  21,  1892,  at  5%. 

308.  The  Six  Per  Cent  Method.    . 

The  interest  of  any  sum  of  money,  at  6%  a  year, 

For  2  months,  or  ^  year,  is  .01  of  the  principal. 
For  1  month  is  ^  of  .01  of  the  principal. 
For  6  days,  or  \  month,  is  .001  of  the  principal. 
For  1  day,  is  i  of  .001  of  the  principal. 


INTEREST.  287 

We  then  have  the  following  rules  for  reckoning  interest 
at  6%  : 

I.   Multiply  .01  of  the  principal  by  ^  the  number  of  months. 
II.   Multiply  .001  of  the  principal  by  \  the  number  of  days. 

III.   Multiply  .001  of  the  principal  by  the  number  of  days, 
and  divide  by  6. 

If  the  given  time  is  any  number  of  years  and  months, 
Rule  I.  should  be  used. 

If  it  is  any  number  of  months  and  days,  use  Rule  II.  if 
the  days  can  be  divided  by  6 ;  otherwise,  use  Rule  III. 

EXAMPLES. 
1.   Find  the  interest  of  $  926  for  3  y.  11  mo.  at  6%. 
$9.26    =  .01  of  the  prin. 


231  =  4  the  no.  of  months 


We  find  .01   of  the  prin- 
2"  vi.^^  y±yj.  v^j.  ixxv^xxuiio.     ^.^^^  ^^  jj^Qyjj^g  ^^^  decimal 

27  78  point  two  places  to  the  left. 
^oK  2  3  y.  11  mo.  is  47  mo. ;  and 

.  ^^  ^  the  number  of  months  is 

^^"^  23^. 


217.61,  Ans. 

2.   Find  the  interest  of  f  347  for  3  mo.  18  d.  at  6%. 

0.347  =  .001  of  the  prin. 

HO       ,1.1,              £  J  We  find  .001  of  the  principal  by 

18  =  4-  the  no.  of  days.  .      ,,      ,    .     i       .  ^  ^. 

o                            "^  movmg  the    decimal  pomt    three 


2  776  places  to  the  left. 

3  47  3  mo.  18  d.  is  108  d.  ;  and  ^  the 

number  of  days  is  18. 


$6.2^6  =  $6.25,  Ans. 

3.   Find  the  amount  of  $  152.75  for  67  days  at  6%. 


$0.15275  =  .001  of  the  prin. 
67  =  the  no.  of  days. 

$152.75 
1.71 

1  06925 
91650 

$154.46  = 

=  the  amount, 
Ans. 

6)10.23425 
$1,705+  =  the  interest. 

288  ARITHMETIC. 

Find  the  interest  at  6%  of  : 

4.  1^  461  for  5  y.  10  mo.  7.  $  146.03  for  43  d. 

5.  $  71.25  for  2  y.  5  mo.  8.  $  507.57  for  96  d. 

6.  $  338  for  3  mo.  24  d.  9.  ^  229.71  for  6  mo.  5  d. 

10.  ^  687.40  from  June  13, 1889,  to  Jan.  25,  1891. 

11.  $  4174.12  from  April  9,  1892,  to  May  8, 1892. 

Find  the  amount  at  6%  of: 

12.  $  28.64  for  4  y.  9  mo. 

13.  $  329.97  for  2  mo.  6  d. 

14.  $  5165.38  from  Feb.  21,  1889,  to  March  14,  1889. 

15.  $  950.89  from  Nov.  15,  1890,  to  July  26,  1892. 

To  find  the  interest  at  any  other  rate,  by  the  six  per 
cent  method,  multiply  the  interest  at  6%  by  the  fraction 
obtained  by  dividing  the  given  rate  by  6. 

Thus,  to  find  interest  at  3|%,  we  multiply  the  interest  at 

6%  by  ^,  or  f 

In  certain  cases  the  work  may  be  simplified;  thus,  for 
interest 

At  4%,  subtract  from  the  6%  interest  i  of  itself. 
At  41%,  subtract  from  the  6%  interest  \  of  itself. 
At  5%,  subtract  from  the  6%  interest  i  of  itself. 
At  7%,  add  to  the  6%  interest  -i-  of  itself. 
At  8%,  add  to  the  6%  interest  i  of  itself. 

16.  Find  the  interest  of  $  72.30  for  5  mo.  12  d.  at  8%. 


$  0.0723  =  .001  of  the  prin. 
27  =  ^  the  no.  of  days. 


After  finding  the   interest 


H  AAQ  at  6%  to  be  $1.9521,  we  add 

to  this  sum  ^  of  itself,  giving 


3)^1.9521  =  int.  at  6 % .  ^^  g^gS 

.6507 
$2.6028  =  int.  at  S%,  A71S. 


INTEREST. 


239 


Find  the  interest  of : 

17.  $  695  for  4  y.  8  mo.  at  5%. 

18.  ^2926  for  24  d.  at  7%. 

19.  $807.10for  17d.  at3i%. 

20.  $948  from  July  14,  1887,  to  May  21,  1888,  at  3%. 

21.  $472  from  Nov.  22,  1890,  to  Sept.  9,  1892,  at  ^%. 

22.  $  565.35  from  Jan.  26,  1893,  to  March  12,  1893,  at 

Find  the  amount  of  : 

23.  f  113.65  for  6  y.  3  mo.  at  4%. 

24.  $156.30  for  53  d.  at  3^%. 

25.  $  8789  from  April  4,  1891,  to  Nov.  16,  1892,  at  2f  %. 

26.  $325.50  from  Dec.  23,  1889,  to  Feb.  3,  1890,  at  8%. 

309.  Interest  for  Aliquot  Parts  of  Two  Hundred  Months. 

Since  a  sum  of  money,  at  6%  interest,  gains  .01  of  itself 
in  2  months,  it  will  gain  the  whole  of  itself  in  200  months. 

Then  the  interest  at  6%  of  any  sum  of  money  may  be 
found  by  taking  that  fractional  part  of  the  principal  which  the 
given  time  is  of  200  months. 

The  following  table  gives,  for  each  of  the  stated  times, 
the  corresponding  fractional  part  of  the  principal : 


100    mo. 

h 

25    mo. 

i 

8Jmo. 

^V 

2  mo. 

T^O 

66 1  mo. 

i 

20   mo. 

tV 

8    mo. 

^V 

J.  mo. 

200 

50   mo. 

i 

16f  mo. 

tV 

5    mo. 

.V 

15  d. 

¥70 

40    mo. 

i 

12 1  mo. 

tV 

4    mo. 

.V 

12  d. 

sh 

33J  mo. 

h 

10    mo. 

IV 

31  mo. 

^v 

6d. 

Woo 

EXAMPLES. 

1.   Find  the  interest  of  $  436.25  for  1  y.  4  mo.  20  d.  at  6  %. 

1  y.  4  mo.  20  d.  is  16|  mo. 

Then  the  interest  is  ^l  of  1436.25,  or  $36.35,  Ans. 


240  ARITHMETIC. 

Find  the  interest  of : 

2.  $2763.53  for  5  y.  6  mo.  20  d.  at  6%. 

3.  f  503.76  for  15  d.  at  6%. 

4.  $310.09  from  May  22,  1887,  to  June  6,  1888,  at  6%. 

5.  $  417.27  for  3  y.  4  mo.  at  4|-%. 

6.  $378.95  for  8  mo.  10  d.  at  4%. 

7.  $821.34  from  Aug.  28,  1892,  to  Sept.  9,  1892,  at  7%. 

Find  the  amount  of : 

8.  $5946.41  for  2  y.  1  mo.  at  6%. 

9.  $  247.18  from  Dec.  15, 1890,  to  March  25, 1891,  at  5%. 

10.  $  192.63  for  8  mo.  at  3|%. 

11.  $795.80  from  Feb.  9,  1887,  to  Nov.  19, 1889,  at  2i%. 

310.  Given  the  Interest,  Time,  and  Principal  or  Amount, 
to  find  the  Rate. 

1.  At  what  rate   must   $384   be   on   interest  to  yield 
$41.76  in  2  y.  5  mo.  ? 

$3.84       $9.28) $41.76(41  We  find  by  the  six  per  cent 

"1^4 1                         3 J  12  method  that  the  interest  on  $  384 

TF^                           A  f\A  ^^^  2  y.  5  mo,,  at  one  per  cent,  is 

^8  A                               Q9S^~2"  $9.28. 

^^  ^                             ^  ^^  Dividing    the    given    interest, 

1  92  $41.76,  by  this  result,  the   qiio- 

6)55.68  tient  is  ^. 

$9.28  =  int.  at  1%.  Then    the    required     rate     is 

^%,Ans. 

Note.    If  the  Interest,  time,  and  amount  are  given,  the  principal 
may  be  found  by  subtracting  the  interest  from  the  amount. 

EXAMPLES. 

At  what  rate  per  cent  will 

2.  $  480  gain  $  47  in  3  y.  11  mo.  ? 

3.  $  51.20  amount  to  $  52.64  in  9  mo.  ? 


INTEREST.  241 

4.  $  6480  gain  $  72.36  in  2  mo.  7  d.  ? 

5.  $  3780  gain  $  5.88  in  28  d.  ? 

6.  $  744  amount  to  $  764.15  in  10  mo.  ? 

7.  $  216  gain  $  25.74  in  4  y.  4  mo.  ? 

8.  $8064  gain  $415.52  in  8  mo.  25  d.  ? 

9.  $  720  amount  to  $  727.35  in  7  mo.  ? 

10.  $  192  amount  to  $  202.44  in  2  y.  5  mo.  ? 

11.  $4320  gain  $  110.88  in  7  mo.  21  d.  ? 

12.  $960  gain  $160.56  from  April  2,  1883,  to  Oct.  29, 
1888? 

13.  $384  amount  to  $388.76  from  March  7,  1892,  to 
June  22,  1892  ? 

14.  $  1200  amount  to  $  1215.05  from  July  25,  1891,  to 
Dec.  4,  1891  ? 

15.  $4455  gain  $147.51  from  Aug.  12,  1889,  to  Jan.  10, 
1890  ? 

16.  At  what  rate  will  a  sum  of  money  double  itself  in 
13  y.  4  mo.  ? 

17.  At  what  rate  will  a  sum  of  money  double  itself  in 
33  y.  4  mo.  ? 

311.  Given  the  Principal  or  Amount,  Interest,  and  Rate, 
to  find  the  Time. 

1.   How  long  must  $94  be  on  interest  at  6%  to  gain 
$19.27? 
$94 
.06 
$5  64")$  19  27^3  ^    V  '^^^  interest  of  $94  for  one  year  at  6% 
16  92     ^    *     is  $5.64. 
rt  ok  Then  to  gain  $  19.27,  it  will  take  as  many 
=  ^         years  as  $5.64  is  contained  times  in  1 19.27. 

3  y.  5  mo.,  Ans. 


242  ARITHMETIC. 

2.    How  long  must  $  157  be  on  interest  at  41%  to  amount 
to  $  162.20  ? 

$157  $162.20  .736  y.  The    interest    of 

.04^  157.  12  $157  fori  y.  at  4^0/^ 


is  $7. 065. 

Subtracting  $157 
from    $162.20,    the 

$  7.065)  $5.2000  (.736  y.  24.96  d.        ^^^^^     ^^^^^^^^     ^^ 

$  O.20. 


628  $  5.20  8.832  mo. 

785  30 


4  9455 


Dividing  $  5.20  by 


25450         e  o-  ^      A  ^1.065,  the  quotient 

21195  "^^^  '  to  the  third  place  of 

decimals  is  .736  y., 


42550  or  8  mo.  25  d. 

Note.  The  quotient  need  never  be  carried  beyond  the  third  place 
of  decimals. 

To  reduce  .736  y.  to  months  and  days,  we  multiply  it  by  12,  giving 
8.832  mo.,  and  then  multiply  .832  by  30,  giving  24.96  d. 

The  final  result  should  be  expressed  to  the  nearest  day  ;  the  number 
of  days  being  increased  by  1  if  the  decimal  is  .5  or  more  than  .5. 

EXAMPLES. 
Find  the  time  in  which 

3.  $ 46  will  gain  $  11.73  at  6%. 

4.  $250  will  gain  $42.50  at  6%. 

5.  $  348  will  amount  to  $356.70  at  5%. 

6.  $  637.50  will  amount  to  $  794.75  at  8%. 

7.  $936  will  gain  $27.30  at  6%. 

8.  $147.50  will  gain  $2.85  at  2%. 

9.  $353.60  will  amount  to  $398.68  at  7^%. 

10.  $741  will  gain  $22.60  at  6%. 

11.  $258.25  will  amount  to  $270.13  at  6%. 

12.  $  801.50  will  amount  to  $  811.40  at  ^%. 

13.  $  716.84  will  gain  $  8.48  at  6%. 


INTEREST. 


243 


14.  $  95.30  will  amount  to  $  97.84  at  4%. 

15.  $  1816  will  gain  $  6.36  at  6%. 

16.  $809.20  will  gain  $30.24  at  4i%. 

17.  $  264.08  will  amount  to  $ 349.91  at  6%. 

18.  $  56.92  will  gain  $ 0.79  at  2i  %. 

19.  In  what  time  will  a  sum  of  money  double  itself  at 
6%  interest? 

20.  In  what  time  will  a  sum  of  money  double  itself  at 
^</o  interest? 

312.    Given  the  Interest  or  Amount,  Time,  and  Rate,  to 
find  the  Principal. 

^  interest  will  gain  $  6.72  in  3  y. 


1.    What  principal  at 
6  mo.? 

$0.01 
21 


$0.21)  $6.72  ($32,  Ans. 
63 

42 


We  find  the  interest  of  $  1  for  3  y. 
6  mo.  at  6%  by  multiplying  .01  of  $1 
by  ^  the  number  of  months. 

Then,  if  one  dollar  gains  $0.21,  to 
gain  $6.72  will  take  as  many  dollars 
as  $0.21  is  contained  times  in  $6.72. 

Dividing  $6.72  by  $0.21,  the  quo- 
tient  is  32. 


2.   What  principal  at  5%  interest  will  amount  to  $150  in 
10  mo.  18  d.  ? 


$150 
6 

$aooi 

53 

$0,053  x^  =  ^^-^^^l 
6          6 

<8^   ,  $0,265      $6,265 
^6               6 

$  6.265)$  900  ($  143.66,  ^ns. 
6265 
27350 
25060 
22900 
18795 

=  amount  of  $  1  at  5% 

41050 
37590 

34600 


244  ARITHMETIC. 

We  first  find  the  interest  of  $1  for  10  mo.  18  d.  at  6%  by  multi- 
plying .001  of  $  1  by  I  the  number  of  days. 

Multiplying  the  result  by  f,  we  find  the  interest  at  5%  to  be 

L J    and  the  amount  of  $  1   for   the  given  time   and  rate  is 

$1   I  $0.265^  ^^  $6.265^ 

6      '  6  ^  6  265 

Then,  if  $1  amounts  to  ^-^ — ^  to  amount  to  $150  will  require 

^6  265 
as  many  dollars  as  ^-^^ —  is  contained  times  in  $  150. 
6 

To  divide  $  150  by  Mi^^S^  ^^  multiply  $  150  by  6,  and  divide  the 
6 

product  by  |;  6.265 ;  the  result  to  the  nearest  cent  is  $  143.66. 


EXAMPLES. 
What  principal  at  interest 

3.  At  6%  will  gain  $  68.20  in  3  y.  8  mo.  ? 

4.  At  8%  will  gain  $  7.91  in  6  mo.  ? 

5.  At  6%  will  amount  to  f  251.86  in  4  y.  9  mo.  ? 

6.  At  6%  will  gain  $  7 AS  in  5  mo.  ? 

7.  At  4%  will  amount  to  f  215.75  in  5  y.  2  mo.  ? 

8.  At  6%  will  gain  $44.03  in  1  y.  1  mo.  12  d.? 

9.  At  7%  will  amount  to  $99.94  in  2  y.  3  mo.  ? 

10.  At  6%  will  gain  $54.98  in  11  mo.  18  d.  ? 

11.  At  5%  will  amount  to  $  77  in  9  mo.  6  d.  ? 

12.  At  6%  will  amount  to  $  307.31  in  36  d.  ? 

13.  At  6%  will  gain  $  10.59  in  6  mo.  15  d.  ? 

14.  At  6%  will  amount  to  $252.04  in  27  d.  ? 

15.  At  2%  will  gain  $  1.60  in  84  d.  ? 

16.  At  4|%  will  gain  $  1.65  in  10  mo.  24  d.  ? 

17.  At  6%  will  amount  to  $594.36  in  7  mo.  11  d.  ? 

18.  At  li%  will  gain  $  11.70  in  2  mo.  7  d.  ? 

19.  At  21%  will  gain  $5.61  in  4  mo.  17  d.  ? 

20.  At  3^%  will  amount  to  $138.92  in  7  mo.  5  d.  ? 


INTEREST.  245 


EXACT  INTEREST. 

313.  Exact  Interest  is  interest  calculated  for  parts  of  a 
year  by  considering  the  actual  number  of  days  in  the  given 
time. 

It  is  used  by  the  United  States  government  on  its  securi- 
ties, and  in  some  business  transactions. 

314.  In  computing  exact  interest,  the  year  is  taken  as 
365  days  instead  of  360  days  as  in  the  common  method. 

Let  it  be  required,  for  example,  to  lind  the  interest  of  a 
sum  of  money  for  43  days. 

By  the  common  method,  the  interest  is  ■^-^,  and  by  the 
exact  method  ^-^-^,  of  the  interest  for  one  year. 

Now  fi,  =  ^\  X  If f  =  ^  X  H- 

That  is,  the  exact  interest  is  ^  of  the  common  interest. 
We  then  have  the  following  rule  for  computing  exact 
interest  for  parts  of  a  year : 

Subtract  from  the  common  interest  -^  of  itself. 

Note  1.  It  must  be  clearly  understood  that  this  rule  does  not 
apply  to  times  greater  than  one  year ;  thus,  to  find  exact  interest  for  2 
y.  3  mo.  18  d.,  we  should  find  the  common  interest  for  2  y.,  and  then 
the  exact  interest  for  3  mo.  18  d.  by  the  above  rule,  and  add  the  results. 

EXAMPLES. 

1.  Find  the  exact  interest  of  $2500  for  2  y.  3  mo.  18  d. 
at  6%. 

We  find  the  com- 
mon interest  of 
$2500  for  2  y.  by 
multiplying  the 
principal  by  .12. 

We  then  find  the 
common  interest 
of  $  2500  for  3  mo. 


$2500 
.12 

$2.50            $45.00 
18                   .62 

$300  = 
int.  for  2  y. 

20  00               44.38 
25  0               300. 
73)  $  45.00  (.62   $  344.38,  Ans. 
43  8 

120 

the  result  is  $  45.00. 


246  ARITHMETIC. 

Dividing  this  by  73,  the  quotient  to  the  nearest  cent  is  $0.62. 

Subtracting  $0.62  from  $45.00,  the  exact  interest  for  3  mo.  18  d.  is 
$  44.38  ;  and  adding  to  this  the  interest  for  2  y.,  $300,  the  final  result 
is  $344.38. 

Find  the  exact  interest 

2.  Of  $942.50  for  3  y.  4  mo.  at  6%. 

3.  Of  $248.25  for  2  y.  9  mo.  at  4%. 

4.  Of  $539.84  for  10  mo.  at  7%. 

5.  Of  $7162  for  7  mo.  at  21%. 

6.  Of  $1930  from  May  5,  1888,  to  Nov.  21,  1888,  at  5%. 

7.  Of  $804  from  Oct.  29,  1889,  to  July  2,  1890,  at  4i%. 

8.  Of  $665.40  from  April  9,  1892,  to  March  22,  1893,  at 
6%. 

9.  Of  $  3515  from  Aug.  16, 1891,  to  Dec.  8, 1891,  at  31%. 

Note  2.  To  compute  the  times  in  Exs.  6  to  9,  the  exact  number 
of  days  between  the  given  dates  must  be  found.     See  Art.  165. 

PROMISSORY   NOTES. 

315.  A  Promissory  Note,  or  simply  a  Note,  is  a  written 
promise  to  pay  to  a  specified  person  a  specified  sum  of 
money. 

316.  A  Demand  Note  is  one  which  is  due  on  demand. 

If  the  words  "  with  interest "  are  added,  it  draws  interest 
from  the  date  of  the  note. 

FORM  OP  A  DEMAND   NOTE. 

$  215^^.  Boston,  Feb.  18,  1892. 

On  demand,  I  promise  to  pay  to  Edward  WiUia7ns  two  hun- 
dred and  fifteen  -f-^-^  dollars,  with  interest  at  Q^fo- 

Value  received.  rr         r, 

Henry  Davis. 

Note.     A  note  must  always  contain  the  words  "  value  received." 


INTEREST.  247 

317.  A  Time  Note  is  one  which,  is  due  at  the  expiration 
of  a  specified  time  after  the  date  of  the  note. 

FORM  OP  A   TIME  NOTE. 
$  500j%%.  New  York,  July  1,  1889. 

Tliree  months  after  date,  tve  promise  to  pay  to  Martin  Pratt, 
or  order,  Jive  hundred  dollars. 

Value  received.  A.  F.  Stearns  &  Co. 

318.  The  Maker  of  the  note  is  the  person  who  signs  it. 
The  Payee  is  the  person  to  whom  it  is  payable. 

The  Holder  of  a  note  is  the  person  who  owns  it. 

An  Indorser  is  a  person  who  writes  his  name  upon  the 
back  of  a  note,  thereby  becoming  responsible  for  its  pay- 
ment in  case  the  maker  fails  to  pay  it  when  due. 

The  Face  of  a  note  is  the  sum  named  in  it. 

319.  If  the  words  "or  order"  or  "or  bearer"  are  written 
after  the  payee's  name,  the  note  may  be  sold  or  transferred; 
otherwise,  only  the  payee  can  collect  the  amount  due. 

A  note  that  can  be  sold  or  transferred  is  called  Negotiable. 

Note.  The  law  of  Pennsylvania  requires  the  addition  of  the  words 
"without  defalcation"  to  "value  received,"  in  a  negotiable  note  ;  in 
New  Jersey,  the  words  ' '  without  defalcation  or  discount ' '  must  be 
added. 

320.  A  time  note  Matures,  or  becomes  legally  payable, 
three  days  after  the  expiration  of  the  time  specified  in  the 
note. 

These  three  days  are  called  Days  of  Grace. 

The  date  of  maturity  of  a  note  is  usually  indicated  by 
writing  the  date  when  nominally  due  and  the  date  of 
maturity  with  a  line  between  them ;  thus,  Aug.  Yu,  1892. 

If  a  note  is  payable  in  a  certain  number  of  days  after 
date,  the  date  of  maturity  is  found  by  counting  forward 
from  the  date  of  the  note  the  specified  number  of  days,  and 
adding  three  days  of  grace. 


248  ARITHMETIC. 

Thus,  if  a  note  is  payable  60  days  after  Aug.  13,  the  date 
of  maturity  is  Oct.  ^/ij. 

If  a  note  is  payable  in  a  certain  number  of  months  after 
date,  calendar  months  are  understood ;  and  the  note  is  nomi- 
nally due  on  the  corresponding  day  of  the  month,  or  on  the 
last  day  of  the  month  when  there  is  no  corresponding  day. 

Thus,  if  a  note  is  payable  3  months  after  Dec.  31,  1892, 
the  date  of  maturity  is  ^"'^^VAprusj  1^93;  but  if  it  is  paya- 
ble 2  months  after  Dec.  31,  1892,  the  date  of  maturity  is 
^^^' ^Marchsj  1893,  since  February  has  no  31st  day. 

If  a  note  falls  due  on  Sunday  or  on  a  legal  holiday,  it  is 
payable  on  the  next  preceding  business  day. 

If  a  note  is  not  paid  when  due,  a  written  notice,  called  a 
Protest,  is  sent  by  a  notary  public  to  the  indorsers. 

If  not  sent  on  the  last  day  of  grace,  the  liability  of  the 
indorsers  ceases. 

PARTIAL  PAYMENTS. 

321.  Partial  Payments  are  payments  in  part  of  a  note  or 
debt. 

Indorsements  are  records  of  the  partial  payments,  with 
their  dates,  made  on  the  back  of  the  note. 

322.  When  partial  payments  have  been  made  on  an 
interest-hearing  note  or  other  obligation,  various  methods 
are  employed  by  business  men  to  determine  the  amount  due 
at  the  final  settlement. 

323.  The  Merchants'  Rule. 

If  the  final  settlement  is  made  within  a  year  from  the  date 
of  the  note,  it  is  usual  to  employ  the  following,  known  as 

THE    MERCHANTS'    RULE. 

Find  the  amount  of  the  face  of  the  note  from  its  date  to  the 
date  of  settlement  ;  also,  the  amount  of  each  partial  payment 
from  its  date  to  the  date  of  settlement. 


a 

"  ^200 

ii 

"  $150 

a 

"  $300 

INTEREST.  249 

Subtract  the  sum  of  the  amounts  of  the  partial  paymeiits 
from  the  amourit  of  the  face  of  the  note. 

1.  What  amount  is  due  Dec.  8,  1892,  on  a  note  for  $850, 
dated  Jan.  2, 1892,  and  bearing  interest  at  6%,  on  which  the 
following  indorsements  have  been  made:  March  18,  1892, 
$  200 ;  May  5,  1892,  $  150  ;  Aug.  22,  1892,  $  300  ? 

Amount  of  $  850  for  11  mo.    6  d.  $897.60 

8  mo.  20  d.  $  208.67 
7  mo.  3d.  155.33 
3  mo.  16  d.      305.30      669.30 

$  228.30,  xl^is. 

The  amount  of  the  face  of  the  note  from  Jan.  2  to  Dec.  8,  or  11  mo. 
6d.,  at  6%,  is  #897.60. 

The  amount  of  the  first  partial  payment  from  March  18  to  Dec.  8, 
or  8  mo.  20  d.,  is  $208.67. 

The  amount  of  the  second  partial  payment  from  May  5  to  Dec.  8,  or 
7  mo.  3d.,  is  1 155.33. 

The  amount  of  the  third  partial  payment  from  Aug.  22  to  Dec.  8, 
or  3  mo.  16  d.,  is  $305.30. 

The  sum  of  the  amounts  of  the  partial  payments  is  $  669.30  ;  which, 
subtracted  from  the  amount  of  the  face  of  the  note,  leaves  $  228.30. 

EXAMPLES. 

2.  What  amount  is  due  Sept.  20,  1891,  on  a  note  for 
$800,  dated  March  2,  1891,  and  bearing  interest  at  6%,  on 
which  the  following  indorsements  have  been  made  :  April 
30,  1891,  $  200 ;  July  13, 1891,  $  300  ? 

3.  What  amount  is  due  Dec.  12,  1890,  on  a  note  for 
$960,  dated  Jan.  19,  1890,  and  bearing  interest  at  6%,  on 
which  the  following  indorsements  have  been  made  :  March 
27, 1890,  $  180  ;  July  12, 1890,  $  320  ;  Oct.  29, 1890,  $  250  ? 

4.  A  note  for  $2900,  dated  Aug.  28,  1891,  and  bearing 
interest  at  4^%,  had  indorsements  as  follows :  Nov.  15, 1891, 
$800;  Jan.  16,  1892,  $450;  Feb.  27,  1892,  $1100.  How 
much  was  due  May  3,  1892  ? 


260  ARITHMETIC. 

5.  On  a  note  for  $  780,  given  May  18,  1890,  and  drawing 
5%  interest,  three  payments  were  made:  June  10,  1890, 
^330;  Sept.  3,  1890,  $290;  Oct.  17,  1890,  $80.  What  was 
due  April  27,  1891  ? 

6.  On  a  note  for  $2000,  dated  Sept.  3,  1891,  and  bearing 
interest  at  6%,  the  following  indorsements  were  made  : 
Nov.  28,  1891,  $300;  Jan.  20,  1892,  $250;  March  12,1892, 
$725;  May  18,  1892,  $420.  How  much  was  due  July  18, 
1892? 

7.  A  note  for  $  698,  dated  Jan.  24,  1892,  is  indorsed  as 
follows:  Feb.  17,  1892,  $115;  Aug.  5,  1892,  $82;  Aug.  18, 
1892,  $129;  Oct.  11,  1892,  $213.  At  6%  interest,  what  is 
due  Nov.  5,  1892  ? 

8.  On  a  note  for  $  3150,  dated  Nov.  1,  1889,  and  bearing 
interest  at  4%,  the  following  payments  were  made :  Dec. 
27, 1889,  $  1080 ;  May  15, 1890,  $  540 ;  June  21,  1890,  $  310 ; 
Sept.  22,  1890,  $  770.     How  much  was  due  Oct.  21,  1890  ? 

324.  The  United  States  Rule. 

If  the  final  settlement  is  made  more  than  a  year  after  the 
date  of  the  7iote,  the  amount  due  is  found  by  the  following 
rule,  adopted  by  the  Supreme  Court  of  the  United  States, 
and  known  as 

THE  UNITED   STATES  RULE. 

Find  the  amount  of  the  principal  to  the  time  when  the  pay- 
ment, or  the  sum  of  the  payments,  equals  or  exceeds  the  interest 
due. 

From  this  amount  subtract  the  payment,  or  the  sum  of  the 
payments;  regard  the  remainder  as  a  new  principal,  and  pro- 
ceed as  before  to  the  time  of  settlement. 

1.  What  amount  is  due  Jan.  5,  1892,  on  a  note  for  $  750, 
dated  Oct.  13,  1889,  and  bearing  interest  at  6%,  on  which 
the  following  indorsements  have  been  made :  Dec.  28,  1889, 
$325;  Aug.  7,  3890,  $10;  July  1,  1891,  $135? 


INTEREST.  251 

Principal,  $750.00 

Int.  Oct.  13, 1889,  to  Dec.  28, 1889,  2  mo.  15  d.,  9.38 

Amount,  $759.38 

1st  partial  payment,  325.00 

New  Principal,  $434.38 

Int.  Dec.  28, 1889,  to  Aug.  7, 1890,  7  mo.  10  d.,  15.93 
Int.  Aug.  7, 1890,  to  July  1, 1891, 10  mo.  24  d.,  23.46 
Amount,  '  $473.77 

Sum  of  2d  and  3d  partial  payments,  145.00 

New  Principal,  $  328.77 

Int.  July  1,  1891,  to  Jan.  5,  1892,  6  mo.  4  d.,  10.08 

Amount  due  Jan.  5,  1892,  $  338.85 

A71S. 

The  amount  of  $750  from  Oct.  13,  1889,  to  Dec.  28,  1880,  or  2  mo. 
15  d.,  at  6%,  is  $759.38. 

Subtracting  from  this  the  1st  partial  payment,  the  new  principal  is 
$434.38. 

The  interest  of  $  434.38  from  Dec.  28,  1889,  to  Aug.  7,  1890,  or  7 
mo.  10  d.,  is  $15.93,  which  is  greater  than  the  2d  partial  payment; 
we  then  compute  interest  on  the  principal  $  434.38  to  the  date  of  the 
next  partial  payment. 

The  interest  of  $434.38  from  Aug.  7,  1890,  to  July  1,  1891,  or  10 
mo.  24  d.,  is  $  23.46  ;  adding  $434.38,  $  15.93,  and  $23.46,  the  amount 
is  $473.77. 

Subtracting  from  this  the  sum  of  the  2d  and  3d  partial  payments, 
or  $  145,  the  new  principal  is  $  328.77. 

The  amount  of  $328.77  from  July  1,  1891,  to  Jan.  5,  1892,  or  6  mo. 
4  d.,  is  $338.85. 

EXAMPLES. 

2.  What  amount  is  due  July  7, 1892,  on  a  note  for  $  395, 
dated  Jan.  27,  1890,  and  bearing  interest  at  6%,  on  which 
the  following  indorsements  have  been  made :  March  5, 1891, 
$125;  Oct.  25,  1891,  $10. 

3.  On  Nov.  9, 1889, 1  gave  my  note,  on  demand,  for  $  630, 
with  interest  at  5%.  On  Aug.  19,  1890,  I  paid  $20;  and 
on  Nov.  6,  1890,  $  400.     What  was  due  Sept.  6,  1891  ? 


252  ARITHMETIC. 

4.  On  a  note  for  $  1500,  dated  Feb.  22, 1887,  and  drawing 
interest  at  6%,  there  was  paid,  Feb.  27, 1888,  ^250 ;  July  4, 

1889,  $55;  and  Feb.  4,  1890,  ^15  740.     What  was  due  May  4, 
1891? 

5.  A  note  for  $  2120,  dated  Aug.  17,  1891,  and  drawing 
6%  interest,  is  indorsed  as  follows:  April  29,  1892,  $75; 
July  22,  1892,  $880;  Nov.  22,  1892,  $545.  What  is  due 
Dec.  13,  1892  ? 

6.  A  note  for  $  540,  dated  April  6,  1886,  and  bearing 
interest  at  4%,  had  indorsements  as  follows  :  April  30, 1886, 
$100;  Nov.  20,  1886,  $220;  Aug.  23,  1888,  $15.  How 
much  was  due  Nov.  19,  1889  ? 

7.  On  a  note  for  $  806,  given  May  15,  1888,  and  bearing 
interest  at  6%,  four  payments  were  made  :  Feb.  3,  1889, 
$  30 ;  July  28,  1889,  $  290 ;  June  24,  1890,  $  24 ;  Oct.  3, 

1890,  $  156.     What  was  due  Feb.  27, 1892  ? 

8.  A  note  for  $  480,  dated  March  6,  1885,, is  indorsed  as 
follows:  Aug.  29,  1885,  $15;  Sept.  22,  1885,  $170;  Sept. 
12,  1886,  $  20.  At  7%  interest,  how  much  is  due  April  14, 
1887? 

9.  On  a  note  for  $  925,  dated  Dec.  2,  1887,  and  bearing 
6%  interest,  the  following  payments  were  made  :  June  17, 
1888,  $110;  Sept.  23,  1889,  $30;  Nov.  1,  1890,  $25; 
Oct.  17, 1891,  $  460 ;  and  March  27, 1892,  $  175.  How  much 
was  due  Aug.  21,  1892  ? 

COMPOUND   INTEREST. 

325.  Compoiind  Interest  is  interest  reckoned  both  on  the 
principal,  and  on  the  unpaid  interest  after  it  becomes  due, 
which  is  added  to  the  principal  at  regular  intervals. 

The  unpaid  interest  may  be  added  to  the  principal,  or 
compounded,  at  the  end  of  each  year,  half-year,  quarter,  or 
any  other  period  of  time,  according  to  agreement. 


INTEREST. 


253 


1st  Principal, 

^  500.00 

Int.  for  1  y., 

30.00 

2d  Principal, 

f  530.00 

Int.  for  1  y., 

31.80 

3d  Principal, 

$  561.80 

Int.  for  7  mo.  12  d., 

20.79 

Compound  amt., 

$  582.59 

Griven  principal. 

500.00 

Compound  int.. 

$  82.59,  Ans. 

326.  1.  What  will  be  the  compound  interest  and  amount 
of  $  500  for  2  y.  7  mo.  12  d.,  at  6%,  the  interest  being  com- 
pounded annually  ? 

The  interest  of  $500 
for  1  year  is  f  30;  which, 
added  to  $  500,  gives  $  530 
as  the  principal  for  the  2d 
year. 

The  interest  of  $530 
for  1  year  is  $  31.80;  which, 
added  to  $530,  gives 
$561.80  as  the  principal 
for  the  7  mo.  12  d. 

The  interest  of  $  561.80 
for  7  mo.  12  d.  is  $20.79  ;  which,  added  to  $561.80,  gives  $582.59  as 
the  compound  amount  of  $  500  for  2  y.  7  mo.  12  d. 

Subtracting  the  given  principal  from  this,  the  compound  interest  is 
$82.59. 

From  the  above  example,  we  derive  the  following 

RULE    FOR    COMPOUND   INTEREST. 

Find  the  amount  of  the  given  principal  for  the  first  period 
of  time. 

Using  this  amount  as  a  new  principal,  find  its  amount  for 
the  second  period  of  time  ;  continuing  in  this  way  until  the 
entire  time  has  been  taken. 

The  last  amount,  less  the  given  principal,  will  be  the  com- 
pound interest  required. 

Note.  If  the  interest  is  compounded  semi-annually  the  rate  must 
be  taken  as  one-half  of  the  yearly  rate  ;  and  if  compounded  quarterly, 
the  rate  must  be  taken  as  one-fourth  of  the  yearly  rate. 

The  interest  is  taken  as  compounded  annually,  in  the  following 
examples,  if  nothing  is  said  to  the  contrary. 


EXAMPLES. 
2.   What  is  the  compound  interest  of  ^  820  for  3  y.,  at 


254 


ARITHMETIC. 


3.  What  is  the  amount  of  $  200  for  4  y.  5  ino.,  at  6% 
compound  interest  ? 

4.  What   is  the  compound  interest  of   $  575  for  5  y., 
at  2%  ? 

5.  What  is  the  compound  interest  of  $  1050  for  2  y. 
6  mo.,  at  6%,  interest  being  compounded  semi-annually  ? 

6.  What  is  the  amount  of  $  436  for  1  y.  9  mo.  24  d.,  at 
3i%  compound  interest  ? 

7.  What  is  the  compound  interest  of  $  3960  for  1  y.  6  mo., 
at  3%,  interest  being  compounded  semi-annually? 

8.  What  is  the  amount  of  f  758  for  11  mo.  10  d.,  at  6%, 
interest  being  compounded  quarterly  ? 

9.  What  is  the  compound  interest  of  $  6000  for  1  y. 
8  mo.,  at  6%,  interest  being  compounded  every  4  months  ? 

10.  What  is  the  compound  interest  of  $  940  for  3  y.  2  mo. 
20  d.,  at  4^%  ? 

11.  What  is  the  amount  of   $1700  for  9  mo.,  at   7%, 
interest  being  compounded  quarterly  ? 


327.  Given  the  Compound  Interest  or  Amount,  Time,  and 
Rate,  to  find  the  Principal. 

1.  What  sum  of  money,  at  3%  compound  interest,  will 
amount  to  $  500  in  2  y.  8  mo.,  interest  being  compounded 
annually  ? 


2d  principal, 
Int.  for  1  y., 
3d  principal, 
Int.  for  8  mo.. 
Compound  Amt., 


$  1.03      $  1.082118)$  500.0000($  462.06, 


.0309 


1.0609 

.021218 
1.082118 


432  8472 
67  15280 
64  92708 
2  225720 
2 164236 
6148400 


Ans. 


INTEREST.  255 

The  amount  of  $1  for  2  y.  8  mo.,  at  3%  compound  interest,  is 
$1.082118. 

Then,  if,$l  amounts  to  $1.082118,  to  amount  to  $500  will  require 
as  many  dollars  as  $1.082118  is  contained  times  in  $500 ;  the  result 
to  the  nearest  cent  is  $462.06. 

Note.  If  the  compound  interest  is  given,  divide  the  given  interest- 
by  the  compound  interest  of  $  1  for  the  given  time  and  rate. 


EXAMPLES. 

2.  What  sum  of  money,  at  6%  compound  interest,  will 
produce  $  70  in  2  years  ? 

3.  What  sum  of  money,  at  4%  interest,  will  amount  to 
$1800  in  1  y.  6  mo.,  the  interest  being  compounded  semi- 
annually ? 

4.  What  principal,  at  6%  compound  interest,  will  gain 
$  105  in  3  y.  9  mo.  ? 

5.  What  sum  of  money,  at  5%  compound  interest,  will 
gain  $  125.38  in  3  y.  8  mo.? 

6.  What  principal,  at  6%,  compound  interest,  will  gain 
$  36  in  9  months,  interest  being  compounded  quarterly  ? 

7.  What  principal,  at  6%  compound  interest,  will  amount 
to  $  725  in  8  mo.,  interest  being  compounded  every  2  months  ? 

8.  What  sum  of  money,  at  6%  compound  interest,  will 
gain  $  80  in  1  y.  8  mo.,  interest  being  compounded  semi- 
annually ? 

328.   Compound  Interest  Tables. 

In  practice,  the  solution  of  problems  in  compound  interest 
may  be  abridged  by  means  of  the  following  tables,  which 
give  the  amount  of  $  1  at  compound  interest,  for  any  num- 
ber of  years  from  1  to  20  inclusive,  at  1^-,  2,  2^,  3,  31  4, 
5,  6,  7,  8,  9,  and  10  per  cent;  interest  being  compounded 
annually. 


256 


ARITHMETIC. 
COMPOUND  INTEREST  TABLES. 


Yrs. 

1^  per  cent. 

2  per  cent. 

2^  per  cent. 

3  per  cent. 

3^  per  cent. 

4  per  cent. 

1 

1.015000 

1.020000 

1.025000 

1.030000 

1.035000 

1.O40000 

2 

1.030225 

1.040400 

1.050625 

1.060900 

1.071225 

1.081600 

3 

1.045678 

1.061208 

1.076891 

1.092727 

1.108718 

1.124864 

4 

1.061364 

1.082432 

1.103813 

1.125509 

1.147523 

1.169859 

5 

1.077284 

1.101081 

1.131408 

1.159274 

1.187686 

1.216653 

6 

1.093443 

1.126162 

1.159693 

1.194052 

1.229255 

1.265319 

7 

1.109845 

1.148686 

1.188686 

1.229874 

1.272279 

1.315932 

8 

1.126493 

1.171660 

1.218403 

1.26(>770 

1.316809 

1.368569 

9 

1.143390 

1.195093 

1.248863 

1.304773 

1.362897 

1.423312 

10 

1.160541 

1.218994 

1.280085 

1.343916 

1.410599 

1.480244 

11 

1.177949 

1.243374 

1.312087 

1.384234 

1.459970 

1.539454 

12 

1.195618 

1.268242 

1.344889 

1.425761 

1.511069 

1.601032 

13 

1.213552 

1.293607 

1.378511 

1.468534 

1.563956 

1.665074 

14 

1.231756 

1.319479 

1.412974 

1.512590 

1.618695 

1.731676 

15 

1.250232 

1.345868 

1.448298 

1.557967 

1.675349 

1.800944 

16 

1.268985 

1.372786 

1.484606 

1.604706 

1.733986 

1.872981 

17 

1.288020 

1.400241 

1.521618 

1.652848 

1.794676 

1.947901 

18 

1.307341 

1.428246 

1.559659 

1.702433 

1.857489 

2.025817 

19 

1.326951 

1.456811 

1.598650 

1.753506 

1.922501 

2.10(5849 

20 

1.346855 

1.485947 

1.638616 

1.806111 

1.989789 

2.191123 

Yre. 

5  per  cent. 

6  per  cent. 

7  per  cent. 

8  per  cent. 

9  per  cent. 

10  per  cent. 

1 

1.050000 

1.060000 

♦ 

1.070000 

1.080000 

1.090000 

1.100000 

2 

1.102500 

1.123600 

1.144900 

1.166400 

1.188100 

1.210000 

3 

1.157625 

1.191016 

1.225043 

1.25i)712 

1.295029 

1.331000 

4 

1.215506 

1.262477 

1.310796 

1.360489 

1.411582 

1.464100 

5 

1.276282 

1.338226 

1.402552 

1.469328 

1.538624 

1.610510 

6 

1.340096 

1.418519 

1.500730 

1.586874 

1.677100 

1.771561 

7 

1.407100 

1.503630 

1.605781 

1.713824 

1.828039 

1.948717 

8 

1.477455 

1.593848 

1.718186 

1.850930 

1.992563 

2.143589 

9 

1.551328 

1.689479 

1.838459 

1.999005 

2.171893 

2.357948 

10 

1.628895 

1.790848 

1.967151 

2.158925 

2.367364 

2.693742 

11 

1.710339 

1.898299 

2.104852 

2.331639 

2.580426 

2.853117 

12 

1.795856 

2.012197 

2.252192 

2.518170 

2.812665 

3.138428 

13 

1.885649 

2.132928 

2.409845 

2.719624 

3.065805 

3.452271 

14 

1.979932 

2.260fK)4 

2.578534 

2.937194 

3.341727 

3.797498 

15 

.2.078928 

2.396558 

2.759031 

3.172169 

3.642482 

4.177248 

16 

2.182875 

2.540352 

2.952164 

3.425943 

3.970306 

4.594973 

17 

2.292018 

2.692773 

3.158815 

3.700018 

4.327633 

5.064470 

18 

2.406619 

2.854339 

3.379932 

3.996019 

4.717120 

5.659917 

19 

2.526950 

3.025600 

3.616527 

4.315701 

5.141661 

6.116909 

20 

2.653298 

3.207136 

3.869684 

4.660957 

5.604411 

6.727600 

INTEREST.  257 

1.  What  is  the  compound  interest  of  $  400  for  15  y.  6  mo., 
at  6%,  interest  being  compounded  annually  ? 

Amount  of  $1  for  15  y.  at  6%,  $2.396558 

400 

Amount  of  $  400  for  15  y.,  $  958.6232 

Interest  of  $  958.6232  for  6  mo.,       28.759 
Compound  Amount,  $  987.38 

400.00 
Compound  Interest,  $  587.38,  Ans. 

Note  1.  If  the  given  time  extends  beyond  the  limits  of  the  table, 
find  the  amount  for  any  convenient  length  of  time,  and  then  use  this 
amount  for  a  new  principal. 

Note  2.  If  the  interest  is  compounded  semi-annually,  take  one- 
half  the  given  rate,  and  ticice  the  given  time. 

Thus,  to  find  the  compound  interest  of  any  sum  of  money  for  8  y. 
at  6  %,  interest  being  compounded  semi-annually,  we  find  by  the  table 
the  compound  interest  of  the  sum  for  16  y.  at  3  %. 

If  the  interest  is  compounded  quarterly,  take  one-fourth  the  given 
rate,  and  four  times  the  given  time. 


EXAMPLES. 

2.  What  is  the  amount  of  $95  for  5  y.  10  mo.,  at  6%, 
interest  being  compounded  annually  ? 

3.  What  is  the  amount  of  $1250  for  3  y.  9  mo.,  at  6%, 
interest  being  compounded  quarterly  ? 

4.  What  is  the  compound  interest  of  $  800  for  8  y.  7  mo., 
at  5%,  interest  being  compounded  semi-annually  ? 

5.  What  is  the  compound  interest  of  $  480  for  11  y.  5  mo. 
10  d.,  at  4%,  interest  being  compounded  annually  ? 

6.  What  principal,  at  5%  compound  interest,  will  amount 
to  $  500  in  13  y.,  interest  being  compounded  annually  ? 

7.  What  sum  of  money,  at  6  %  compound  interest,  will 
gain  $  230  in  9  y.  8  mo.,  interest  being  compounded  semi- 
annually ? 


268  ARITHMETIC. 


ANNUAL  INTEREST. 

329.  If,  when  interest  is  payable  annually  on  a  note  or 
other  obligation,  the  payments  are  not  made  when  due,  the 
amount  due  at  the  time  of  settlement  may  in  certain  cases 
be  found  by  reckoning  simple  interest  on  the  principal,  and  on 
each  annual  interest  after  it  becomes  due. 

This  is  called  Annual  Interest. 

1.  What  amount  is  due  at  the  end  of  3  y.  6  mo.  12  d.,  on 
a  note  for  $500,  with  6%  interest  payable  annually,  on 
which  no  payments  have  been  made  ? 

Principal,  $500.00 

Int.  of  $  500  f or  3  y .  6  mo.  12  d.,  at  6  % ,  106.00 
Int.  of  $  30  for  4  y.  7  mo.  6  d.,  at  6%,  8.28 
Amount  due,  $  614.28,  Ans. 

The  interest  of  the  principal  for  3  y.  6  mo.  12  d.,  is  f  106. 

Each  annual  interest  is  $  30  ;  the  first  draws  interest  for  2  y.  6  mo. 
12  d. ;  the  second  for  1  y.  6  mo.  12  d. ;  and  the  third  for  6  mo.  12  d. 

In  all,  this  is  equivalent  to  $30  drawing  interest  for  4  y.  7  mo.  6  d., 
which  is  $8.28. 

Then  the  amount  due  is  $614.28. 

EXAMPLES. 

2.  What  amount  is  due  at  the  end  of  2  y.  9  mo.  18  d.,  on 
a  note  for  $  900,  with  6%  interest  payable  annually,  on  which 
no  payments  have  been  made  ? 

3  What  amount  is  due  June  27,  1888,  on  a  note  for 
$385,  dated  Feb.  5, 1885,  with  6%  interest  payable  annually, 
on  which  no  payments  have  been  made  ? 

4.  What  is  the  annual  interest  of  $  763  for  2  y.  5  mo. 
10  d.,  at  5%  ? 

5.  What  interest  is  due  on  a  note  for  $6450,  with  4% 
interest  payable  annually,  at  the  end  of  3  y.  8  mo.  24  d.  ? 

6.  What  amount  is  due  Nov.  12, 1892,  on  a  note  for  $  898, 
dated  Nov.  29,  1887,  with  6%  interest  payable  annually,  on 
which  no  payments  have  been  made  ? 


DISCOUNT.  269 

XX.    DISCOUNTo 

TRUE  DISCOUNT. 

330.  Discount  is  a  reduction  made  from  a  debt  that  is 
paid  before  it  becomes  due. 

The  Present  Worth  of  a  sum  of  money  due  at  some  future 
date  without  interest,  is  that  sum  which,  if  put  at  interest 
for  the  given  time,  will  amount  to  the  given  sum. 

The  True  Discount  is  the  difference  between  the  given 
sum  and  its  present  worth ;  that  is,  it  is  the  interest  of  the 
present  worth  for  the  given  time. 

EXAMPLES. 

331.  1.  What  is  the  present  worth  and  true  discount  of 
f  300,  due  1  y.  5  mo.  hence,  at  5%  ? 


f  1.00 


This  is  the  same  as 
05  finding  what  principal 

17       <8insfi  ^^^^   amount  to    $300 

$0.05  X  i^  =  ^^.  in  1   y.   5  mo.   at  b% 

^^  ^^  (Art.  312). 

0.85       f  12.85  ^^  find  the  interest 

"h^2     ^^      To      *  of  $1  for  1  y.   5  mo. 

at  5%  by  multiplying 

$300  .05  of  $1  by  II;  the 

12  $300  result  is  ^<>-8^- 


1  + 


12.85)  $  3600  ($  280.16,  present  worth.  12 

oe^n-A  ""^"TrToT  4.         J'  i.        Then  the  amount  of 

2570     $  19.84,  true  discount.  ^^  .     ^^ 

'  $  1  for  the  given  time 

•  i-???9  and  rate  is 


10280 


$0.85  _$  12.85 


$1+-*^^^^^,  or 


2000  7^"^1^'"'~T^ 

1^^^  '  To  divide    $300    by 

7150  $12.85  ,,.  ,     .^ 

,  we  multiply  it 

by  12,  and  divide  the  product  by  $  12.85  ;  the  result  to  the  nearest  cent 
is  $  280.16,  which  is  the  present  worth. 

Subtracting  the  present  worth  from  the  given  sum,  $  300,  the  trii^ 
discount  is  $19.84. 


260  ARITHMETIC. 

EXAMPLES. 
Find  the  present  worth  and  true  discount  of  : 

2.  $400  due  2  y.  11  mo.  hence  at  6%. 

3.  $890  due  6  mo.  hence  at  5%. 

4.  $  725  due  4  y.  10  mo.  hence  at  4|^%. 

5.  $  1730  due  3  y.  4  mo.  hence  at  3i%. 

6.  $682  due  5  mo.  27  d.  hence  at  6%. 

7.  $269.20  due  2  mo.  18  d.  hence  at  3%. 

8.  $2500  due  3  mo.  6  d.  hence  at  7%. 

9.  $950  due  9  mo.  11  d.  hence  at  6%. 

10.  $  135.75  due  1  y.  8  mo.  15  d.  hence  at  4%. 

11.  $347.68  due  7  mo.  20  d.  hence  at  2i%. 

BANK   DISCOUNT. 

332.  Bank  Discount  is  a  sum  of  money  charged  .by  a  bank 
for  the  payment  of  a  negotiable  note  (Art.  319)  before  it 
becomes  due. 

It  is  reckoned  as  the  simple  interest  of  the  face  of  the  note 
from  the  day  of  discount  to  the  day  of  maturity  (Art.  320). 

The  time  from  the  day  of  discount  to  the  day  of  maturity 
is  called  the  Term  of  Discount,  and  the  rate  of  interest  is 
called  the  Rate  of  Discount. 

The  Proceeds  or  Avails  of  a  discounted  note  is  its  face 
less  the  bank  discount. 

333.  If  a  note  is  discounted  on  the  day  of  its  date,  the 
term  of  discount  is  the  time  specified  in  the  note,  plus  three 
days  of  grace  (Art.  320). 

If  a  note  is  due  a  certain  number  of  months  after  date, 
the  term  of  discount  is  found  in  months  and  days  (Art.  165)  ; 
if  it  is  due  a  certain  number  of  days  after  date,  the  term  of 
discount  is  found  in  exact  days. 


n 


DISCOUNT.  261 

Note.  It  will  be  understood,  in  the  examples  of  the  present  chap- 
ter, that  the  note  is  discounted  on  the  day  of  its  date  if  nothing  is  said 
to  the  contrary. 

EXAMPLES. 

334.  1.  Find  the  proceeds  of  a  4-months'  note  for  $485, 
discounted  on  the  day  of  its  date  at  6%. 

The  term  of  discount 

Face  of  note,  $  485.00  '^l"""'  f  '■'i^;'-  f  ^>; 

;  By  Art.  332,  the  bank 

Int.  4  mo.  3  d.,  at  6%,  9.94  discount  is  the  interest 

Proceeds,  $  475.06,  Ans.     of  $  485  for  4  mo.  3  d. , 

at  6%,  which  is  $9.94. 
Subtracting  this  from  the  face  of  the  note,  the  proceeds  is  $475.06. 

If  an  interest-bearing  note  is  discounted,  the  discount  is 
reckoned  on  the  amount  due  at  maturity. 

2.  Find  the  proceeds  of  a  note  for  $  1050,  dated  May  12, 
1892,  payable  90  days  after  date,  and  bearing  interest  at 
4%,  if  discounted  June  23,  1892,  at  5%. 

Face  of  note,  $1050.00 

Interest  of  $  1050  for  93  d.,  at  4%,  10.85 

Amount  due  at  maturity,  $  1060.85 

Interest  of  $  1060.85  for  51  d.,  at  5%,  7.51 

Proceeds,  $  1053.34,  Ans. 

The  day  of  maturity  is  93  d.  after  May  12,  1892,  or  Aug.  13,  1892. 
The  interest  of  $  1050  for  93  d.,  at  4  %,  is  $  10.85  ;  whence  the  amount 

due  at  maturity  is  $  1060.85. 

The  term  of  discount  is  the  exact  number  of  days  from  June  23, 

1892,  to  Aug.  13,  1892,  or  51  d. 

Then  the  bank  discount  is  the  interest  of  $  1060.85  for  51  d.  at  5  % ; 

that  is,  $7.51. 

Subtracting  $  7.51  from  $  1060.85,  the  proceeds  is  $  1053.34. 
Note.    The  term  of  discount  in  Ex.  2  may  be  found  without 

obtaining  the  day  of  maturity,  by  counting  the  exact  number  of  days 

from  May  12,  1892,  to  June  23,  1892,  and  subtracting  the  result  from 

93  days. 

3.  Find  the  proceeds  of  a  3-months'  note  for  ^  500,  dis- 
counted at  65^?. 


262  ARITHMETIC. 

4.  What  is  the  proceeds  of  a  60-day  note  for  ^950, 
discounted  at  5%  ? 

5.  What  is  the  bank  discount  on  a  note  for  ^2000,  due 
90  days  hence,  at  7%  ? 

6.  Find  the  bank  discount  on  a  note  for  $  620,  payable 
in  4  months,  and  discounted  2  mo.  9  d.  after  date,  at  6%. 

7.  How  much  money  should  a  bank  pay  to  the  holder 
of  a  note  for  $  1000,  due  in  30  days,  if  discounted  at  4%  ? 

8.  What  is  the  bank  discount  on  a  note  for  $  700,  dated 
July  13,  1891,  and  payable  90  days  after  date,  if  discounted 
Aug.  25,  1891,  at  5i%? 

9.  Find  the  proceeds  of  a  note  for  $6000,  payable  2 
months  after  date,  and  bearing  interest  at  6%,  if  discounted 
at  5%. 

10.  A  note  for  $425.30,  dated  Oct.  4,  1890,  and  payable 
6  months  after  date,  was  discounted  Jan.  24,  1891,  at  6%. 
What  was  the  proceeds  ? 

11.  What  is  the  proceeds  of  a  60-day  note  for  $800, 
dated  March  16,  1892,  and  bearing  interest  at  4%,  if  dis- 
counted April  22,  1892,  at  M%? 

12.  What  charge  will  be  made  by  a  bank  for  discounting, 
at  4%,  a  note  for  $384.50,  due  75  days  hence? 

13.  A  note  for  $  275,  payable  5  months  after  date,  was 
discounted  at  5^%.     What  was  the  bank  discount  ? 

14.  What  sum  will  be  realized  by  discounting  a  30-day 
note  for  $3000 ;  the  rate  of  discount  being  7%  ? 

15.  Find  the  bank  discount  on  a  note  for  $190,  dated 
Aug.  26,  1890,  payable  90  days  after  date,  with  interest  at 
41%,  and  discounted  Nov.  3,  1890,  at  6%. 

16.  A  15-day  note  for  $  503.70  was  discounted  at  a  bank 
at  41%  ;  how  much  did  the  bank  receive  ? 

17.  A  note  for  $  1100,  dated  Sept.  10,  1892,  and  payable 
6  months  after  date,  with  interest  at  6%,  was  discounted 
Feb.  7,  1893,  at  4^%.     What  was  the  proceeds  ? 


DISCOUNT.  263 

18.  Find  the  bank  discount  on  a  note  for  $  9000,  due  in 
4  months,  the  rate  of  discount  being  6^%. 

19.  A  60-day  note  for  $  750,  dated  June  28,  1889,  was 
discounted  July  22,  1889,  at  5f%.  What  was  the  bank 
discount  ? 

20.  Find  the  proceeds  of  a  note  for  §  280,  dated  May  21, 
1888,  payable  45  days  after  date,  with  interest  at  3J%,  and 
discounted  June  7,  1888,  at  6%. 

21.  A  3-months'  note  for  $  400,  dated  Nov.  19,  1892,  was 
discounted  Jan.  3,  1893,  at  6^%.  What  sum  was  received 
by  the  holder  ? 

22.  How  much  should  a  bank  receive  for  discounting  a 
note  for  f  847.25,  payable  30  days  after  date,  with  interest 
at  5%  ;  the  rate  of  discount  being  4f  %  ? 

335.  To  find  the  Face  of  a  Note  to  yield  a  given  Proceeds. 

1.  What  must  be  the  face  of  a  note,  due  90  days  hence, 
which,  when  discounted  at  4%,  will  yield  $  500? 

^  0.001 
93 

iM93^fM31x?=:iM31  =  int.of$lfor93d.at4%. 
6  2  3  3  ^ 

^  ^  _g0p^$3--$0.031^g2|69^  p^^^^^^3  ^f  ^ ^1  ^^,3 

1500 
3 


$  2.969)  f  1500.00  ($  505.22,  Ans. 
1484  5 
15  500 
14  845 
6550 
5938 


6120 

We  first  find  the  interest  of  $  1  for  93  d.  at  6  %  by  multiplying  .001 
of  $  1  by  93,  and  dividing  the  product  by  6  ;  the  result  is  '^   '       ■ 


264  ARITHMETIC. 

Multiplying  this  by  f ,  the  interest  of  $  1  for  93  d.  at  4%  is  ^^^^. 

3 

^2  969 
Subtracting  this  from  $1,  the  remainder  is  ^^— ^ — ,  which  is  the 

o 

proceeds  o/  a  $  1  note  for  90  days. 

Then  to  yield  $  500,  the  face  of  the  note  must  be  as  many  dollars  as 

iM§2  is  contained  times  in  $  500. 
o 

To  divide  $  500  by  iM^,  we  multiply  $  500  by  3,  and  divide  the 
o 
product  by  $2,969;  the  result  to  the  nearest  cent  is  $505.22,  which 
is  the  face  of  the  note  required. 

EXAMPLES. 

2.  What  must  be  the  face  of  a  note,  due  3  months 
hence,  which,  when  discounted  at  6%,  will  yield  $  600  ? 

3.  The  proceeds  of  a  30-day  note,  discounted  at  5%,  was 
$  340.     What  was  the  face  of  the  note  ? 

4.  Wishing  to  borrow  ^225  at  a  bank,  for  what  sum 
must  my  note  be  drawn  at  60  days  to  obtain  that  amount, 
the  rate  of  discount  being  6%  ? 

5.  For  what  amount  must  a  5-months'  note  be  drawn,  so 
that,  when  discounted  at  7%,  the  proceeds  may  be  $  8000? 

6.  The  holder  of  a  75-day  note  received  $  550  as  pro- 
ceeds, when  the  note  was  discounted  at  4%,  What  was  the 
face  of  the  note  ? 

7.  For  what  sum  must  a  note,  payable  in  6  months,  be 
drawn,  to  yield  $  1500  when  discounted  at  4^%  ? 

8.  If  the  rate  of  discount  is  7%,  what  must  be  the  face 
of  a  note,  payable  2  months  hence,  to  yield  $  425  when 
discounted  ? 

9.  For  what  amount  must  my  note,  due  in  90  days,  be 
drawn,  in  order  that  I  may  receive  $  908.70  when  the  note 
is  discounted  at  5|-%  ? 

10.  A  merchant  received  a  60-day  note  in  payment  for 
goods  sold ;  he  at  once  had  it  discounted  at  3|%,  and  realized 
^  375.     For  what  amount  were  the  goods  sold  ? 


EXCHANGE.  265 


XXI.    EXCHANGE. 

336.  A  Draft  is  a  written  order  from  one  person  to 
another,  directing  him  to  pay  a  specified  sum  of  money 
to  a  third  person. 

The  Drawer  of  a  draft  is  the  person  who  signs  it. 
The  Drawee  is  the  person  to  whom  it  is  addressed. 
The  Payee  is  the  person  to  whom  it  is  payable. 
The  Face  of  a  draft  is  the  sum  named  in  it. 

337.  A  Sight  Draft  is  one  which  is  payable  on  presenta- 
tion to  the  drawee. 

FORM  OP  A   SIGHT  DRAFT, 
f  eOOj^.  Philadelphia,  Feb.  12,  1892. 

At  sight,  pay  to  the  order  of  Henry  F.  Sears  six  hundred 
dollars,  value  received,  and  charge  the  same  to  the  account  of 

E.  B.  Hart&  Co. 

To  Stone  &  Morison,  Cleveland,  Ohio. 

The  above  draft  may  be  supposed  to  have  been  drawn  under  the 
following  circumstances  : 

A  merchant  in  Philadelphia  wishes  to  pay  $  600  to  Henry  F.  Sears, 
in  Cleveland. 

He  proceeds  to  a  banking-firm,  E.  B.  Hart  &  Co.,  who  have  an 
account  with  Stone  &  Morison,  of  Cleveland. 

They  sell  him  a  draft  for  the  above  amount,  which  he  forwards  to 
Mr.  Sears  at  Cleveland. 

When  Mr.  Sears  receives  it,  he  carries  it  to  Stone  &  Morison,  who 
pay  it  on  presentation. 

338.  A  Time  Draft  is  one  which  is  payable  at  the  expi- 
ration of  some  specified  time  after  presentation,  or  after 
the  date  of  the  draft. 

It  is  usual  to  allow  three  days  of  grace. 


266  ARITHMETIC. 

FORM   OP  A   TIME  DRAFT. 

f  845y0^.  •         Chicago,  Nov.  23,  1892. 

Sixty  days  after  date,  pay  to  the  order  of  Charles  H.  Jack- 
son eight  hundred  and  forty-Jive  dollars,  value  received,  and 
charge  the  same  to  the  account  of 

William  Rogers. 
To  the  Erie  National  Bank, 
Buffalo,  NY. 

339.  The  Acceptance  of  a  time  draft  by  the  drawee  is  an 
agreement  on  his  part  to  pay  it. 

To  accept  a  draft,  the  drawee  writes  the  word  "Accepted" 
across  its  face,  with  the  date,  and  his  signature. 

The  draft  is  then  called  an  Acceptance,  and  the  drawee 
an  Acceptor. 

An  acceptance  may  be  negotiated  in  the  same  manner  as 
a  promissory  note ;  if  discounted  at  a  bank,  the  term  of  dis- 
count is  the  time  specified  in  the  draft,  plus  three  days  of 
grace  (Art.  333). 

340.  Exchange  is  the  system  of  making  payments  by 
remitting  drafts. 

341.  Exchange  is  said  to  be  at  a  certain  per  cent  Premium, 
or  Above  Par,  when  a  draft  sells  for  the  specified  per  cent 
more  than  its  face  value. 

Thus,  if  exchange  is  at  1  %  premium,  a  draft  for  $  100 
will  sell  for  $100,  plus  1%  of  $  100,  or  $  101. 

Exchange  is  said  to  be  at  a  certain  per  cent  Discount,  or 
Below  Par,  when  a  draft  sells  for  the  specified  per  cent  less 
than  its  face  value. 

Thus,  if  exchange  is  at  1%  discount,  a  draft  for  $  100 
will  sell  for  $  100,  less  1%  of  f  100,  or  $  99. 

The  Rate  of  Exchange  is  the  per  cent  which  a  draft  costs 
more  or  less  than  its  face  value. 


EXCHANGE.  267 


DOMESTIC  EXCHANGE. 

342.  Domestic  or  Inland  Exchange  is  exchange  between 
persons  m  the  same  country. 

EXAMPLES. 

343.  1.  Find  the  cost  of  a  sight  draft  for  $  452,  when 
exchange  is  at  1^%  premium. 

$452      $452.00 

•Qlj  6.78  1^%  of  $452  is  $6.78. 

4  52        $458.78,  Ans.  Then,  since  the  draft  sells  for  $6.78 

2  26  more  than  its  face,  the  cost  is  $458.78. 

$6.78 

In  finding  the  cost  of  a  time  draft,  the  interest  of  the  face 
of  Me  draft  for  the  specified  time,  plus  three  days  of  grace, 
must  be  deducted  from  the  cost;  for  the  drawer  has  the  use 
of  the  money  for  that  length  of  time  before  the  drawee  pays 
the  draft. 

2.  What  will  be  the  cost  of  a  draft  for  $  1000,  due  30 
days  after  sight,  with  interest  at  5%,  exchange  being  at  f  % 
discount  ? 

«innn  V  ^      ^30.00      ^-.  k^  We  find  fo/^  of  $1000  by 

*  1^.^^  X  -  =        ^       =  *  7.50.     jnuitipiying  .01  of  $  1000   by 

$  1.00        $  1000.00  ^  '  *^^  "^"''^^  '^  ^  ^•^^•. 

oo  7  (lO  Then,  since  exchange  is  at 

a  discount,   a  siaht  draft  for 


6)$^33^  $  992.50  ^  ^^^O  will  cost  $  1000  -  $  7.50, 

6)$5^  4.58  or  $992.50. 

.92  $  987.92,  Ans.  The  interest  of  the  face  of 


$4.58  the  draft  for  33  days,  at  5%, 

is$4.-58. 
Deducting  this  from  the  cost  of  the  sight  draft,  the  required  cost  is 
$987.92. 

3.   Find  the  cost  of  a  sight  draft  for  $500,  when  ex- 
change is  at  1\%  premium. 


268       ^  ARITHMETIC. 

4.  What  will  be  the  cost  of  a  sight  draft  for  $  280,  if 
exchange  is  at  |%  discount  ? 

5.  What  will  be  the  cost  of  a  draft  for  $  8000,  due  60 
days  after  sight,  with  interest  at  4%,  exchange  being  at 
■|%  discount? 

6.  What  must  be  paid  for  a  draft  on  New  York  for  $  472, 
due  30  days  after  sight,  with  interest  at  5%,  if  exchange  is 
at  1  %  premium  ? 

7.  I  wish  to  purchase  a  sight  draft  on  San  Francisco  for 
$1965.  If  exchange  is  at  1^%  discount,  how  much  must  I 
pay  for  it  ? 

8.  A  merchant  purchased  a  draft  for  $  700,  due  90  days 
after  sight,  with  interest  at  4^%,  exchange  being  at  \% 
premium.     What  was  the  cost  ? 

9.  How  much  must  be  paid  for  a  draft  for  $  344,  due 
30  days  after  sight,  with  interest  at  3J%,  exchange  being  at 
■|%  premium  ? 

10.  What  will  be  the  cost  of  a  draft  for  $  632,  due  60 
days  after  sight,  with  interest  at  3^%,  if  exchange  is  at  3f  % 
discount  ? 

11.  How  much  must  be  paid  for  a  draft  on  Cincinnati 
for  $1378,  due  3  months  after  sight,  with  interest  at  41%, 
if  exchange  is  at  a  discount  of  2i%  ? 

344.  To  find  the  Face  of  a  Draft  when  the  Cost  is  given. 

1.  What  is  the  face  of  a  sight  draft  which  can  be  bought 
for  $  248.85,  when  exchange  is  at  1^%  discount  ? 


$1.00 
.011 
100 
25 

$1.00 
.0125 
$  .9875 

$.9875)$ 248.85 ($252,  Ans. 
197  50 
51350 
49  375 

$  .0125 

19750 
1  9750 

EXCHANGE.  269 

li  %  of  $  1  is  $  .0125. 

Subtracting  this  from  $  1,  the  cost  of  a  sight  draft  for  $  1  is  $  .9875. 
-Then  if  the  cost  of  the  given  draft  is  $  248.85,  its  face  will  be  as 
many  dollars  as  $.9875  is  contained  times  in  $248.85. 
The  result  is  $262.00. 

2.  What  will  be  the  face  of  a  draft,  due  60  days  after 
sight,  with  interest  at  6%,  which  can  be  bought  for  $  1000, 
when  exchange  is  at  a  premium  of  f  %  ? 

$  0.01  X  ^  =  ^^  =  $  0.00625 
8  8  -| 


$.001  $1.00625 

63  .0105 


6) $.063  $ 0.99575)$  1000.00($  1004.27,  Ans. 

$  .0105  995  75 

4  25000 
3  98300 


267000 
199150 


678500 

We  find  1%  of  $1  by  multiplying  .01  of  $1  by  | ;  the  result  is 
$.00625. 

Then,  since  exchange  is  at  a  premium,  a  sight  draft  for  $1  will 
cost  $  1  +  $  .00625,  or  $  1.00625. 

The  interest  of  $  1  for  63  days  at  6  %  is  $  .0105. 

Deducting  this  from  $1.00625,  the  cost  of  a  draft  for  $1,  due  60 
days  after  sight,  with  interest  at  6%,  will  be  $0.99575. 

Dividing  the  given  cost  by  this,  the  result  to  the  nearest  cent  is 
$  1004.27. 

3.  What  is  the  face  of  a  sight  draft  which  can  be  bought 
for  $  435.60,  if  exchange  is  at  1%  discount  ? 

4.  A  merchant  bought  a  sight  draft  on  Pittsburg  for 
$  657.72,  when  exchange  was  at  l\fo  premium.  What  was 
the  face  of  the  draft  ? 


270  ARITHMETIC. 

5.  What  will  be  the  face  of  a  draft,  due  30  days  after 
sight,  with  interest  at  6%,  which  can  be  bought  for  $  918, 
when  exchange  is  at  a  premium  of  J%  ? 

6.  A  sight  draft  on  Philadelphia  was  bought  for  $  240, 
when  exchange  was  at  J%  discount.  What  was  the  face 
of  the  draft  ? 

7.  Find  the  face  of  a  draft  on  Detroit,  due  60  days  after 
sight,  with  interest  at  4%,  which  can  be  bought  for  $  760, 
exchange  being  at  1J%  premium. 

8.  I  purchased  a  sight  draft  on  St.  Louis  for  $  189.84, 
when  exchange  was  at  1^%  discount.  What  was  the  face 
of  the  draft  ? 

9.  How  large  a  draft,  due  90  days  after  sight,  with  inter- 
est at  3%,  can  be  bought  for  $  575,  when  exchange  is  at  a 
discount  of  J%  ? 

10.  A  merchant  paid  $  3000  for  a  60-day  draft,  with 
interest  at  5%,  exchange  being  at  If  %  premium.  What 
was  the  face  of  the  draft  ? 

11.  How  large  a  draft  on  Baltimore,  due  one  month  after 
sight,  with  interest  at  2^%,  can  be  purchased  for  f  2380, 
when  exchange  is  at  a  premium  of  1|%  ? 

FOREIGN  EXCHANGE. 

345.  Foreign  Exchange  is  exchange  between  persons  in 
different  countries. 

346.  A  draft,  in  foreign  exchange,  is  usually  called  a 
Bill  of  Exchange. 

A  Set  of  Exchange  is  a  series  of  three  bills,  all  of  the 
same  date  and  tenor,  called  the  First,  Second,  and  Tliird  of 
exchange,  respectively. 

They  are  sent  by  different  mails  to  avoid  the  delay  which 
might  arise  from  the  loss  of  a  single  draft. 

If  any  one  of  the  three  is  paid,  the  others  become  void. 


EXCHANGE.  271 

FORM   OF  A   FOREIGN   BILL  OF   EXCHANGE. 

£200.  Boston,  Dec.  3,  1892. 

At  sight  of  this  First  of  Exchange,  second  and  third  of  the 
same  date  and  tenor  unpaid,  pay  to  the  order  of  George  Lewis 
two  hundred  pounds  sterling,  value  received,  and  charge  the 

same  to  the  account  of 

Kidder^  Peabody,  &  Co. 
To  Messrs.  Baring  Brothers, 
London,  England. 

347.  Exchange  on  Great  Britain  or  Ireland  is  quoted  at 
the  value  of  a  pound  sterling  in  United  States  dollars ; 
exchange  on  Erance,  at  the  value  of  a  dollar  in  francs ; 
exchange  on  Germany,  at  the  value  of  4  reichsmarks  in  cents. 

Thus  the  statement 

"Bankers'  Sterling,  sight,  4.86 J;  Commercial  bills,  60 
days,  4.82i;  Francs,  sight,  5. 16 J;  Reichsmarks,  sight,  95|," 
would  be  interpreted  as  follows : 

Sight  drafts  on  a  bank  or  banker  in  Great  Britain  or 
Ireland,  f  4.86J  to  the  pound  sterling;  drafts  drawn  on 
merchants,  due  60  days  after  sight,  $4.82^  to  the  pound 
sterling ;  sight  drafts  on  France,  5.16^  francs  to  the  dollar ; 
sight  drafts  on  Germany,  95f  cents  per  4  marks. 

Sterling  Exchange  is  exchange  on  Great  Britain  or  Ireland. 

EXAMPLES. 

348.  1.  What  will  be  the  cost  of  a  bill  of  exchange  on 
London  for  £  160  8s.,  when  exchange  is  quoted  at  4.85  ? 

£  160  8s.  =  £  160.4 

4.85  £  160  8s.  is  the  same  as  £  160.4. 

80  20  Since    each    pound    sterling   costs 

1283  2  $4.85,    160.4   pounds   will  cost  160.4 

6416  times  $4.85,  or  f  777.94. 
^777.94.,  Ang. 


272  ARITHMETIC. 

2.  Find  the  cost  of  a  bill  of  exchange  on  Paris  for  632 
francs,  exchange  being  quoted  at  5.16J. 

5.1625)  632. 00  ($122.42,  Ans. 
516  25 

115  750  Since    5.1625    francs    can    be 

103  250  bought  for  $  1,  to  buy  632  francs 

12  5000  ^^^^  ^^^^  ^^  many  dollars  as  5. 1625 

10  S250  '^^  contained  times  in  682. 

o  i7KrtA  '^^^  result  to  the  nearest  cent 

2  06500  •      ''''''■''' 
110000 

3.  How  large  a  draft  on  Berlin  can  be  bought  for  $  100, 
if  exchange  on  Germany  is  quoted  at  95^  ? 

$100 
4 

$.955)$ 400.00(418.85  marks,  Ans. 

^M^  Since     4     marks     cost 

18  00  $  .955,  as  many  marks  can 

9  55  be    bought    for    flOO    as 

8  450  955  is  contained  times  in 

7  640  4  times  $  100,  or  $  400. 
8100 
7640 


4600 


4.  A  merchant  bought  a  bill  of  exchange  on  London  for 
£348,  when  exchange  was  quoted  at  4. 86 J.  What  was  the 
cost? 

5.  Find  the  cost  of  a  draft  on  Berlin  for  848  marks, 
exchange  on  Germany  being  quoted  at  95^. 

6.  How  much  must  be  paid  for  a  bill  of  exchange  on 
Paris  for  3000  francs,  exchange  at  5.16  ? 

7.  How  large  a  draft  on  London  can  be  bought  for 
$190.12,  exchange  at  4.85  ? 

8.  How  large  a  draft  on  Paris  can  be  bought  for  $  5786, 
exchange  at  5.17^  ?  » 


EXCHANGE.  273 

9.   Find  the  cost  of  a  bill  of  exchange  on  Liverpool  for 
£16  5s.,  exchange  at  4.85^. 

10.  If  exchange  on  Germany  is  quoted  at  94|,  how  much 
must  be  paid  for  a  draft  on  Bremen  for  2175  marks  ? 

11.  If  exchange  on  Paris  is  quoted  at  5.181,  how  much 
must  be  paid  for  a  bill  of  exchange  for  725  francs  ? 

12.  A  merchant  paid  $  7085.03  for  a  draft  on  Berlin,  ex- 
change on  Germany  at  96|.  What  was  the  face  of  the 
draft  ? 

13.  How  much  must  be  paid  for  a  bill  of  exchange  on 
Bristol  for  £523  17s.,  exchange  at  4.88i? 

14.  What  is  the  face  of  a  bill  of  exchange  on  Glasgow 
costing  $  858,  exchange  at  4.87|-  ? 

15.  If  exchange  on  France  is  quoted  at  5.19J,  how  large 
a  draft  on  Marseilles  can  be  bought  for  $  946.50  ? 

16.  Find  the  cost  of  a  bill  of  exchange  on  Belfast  for 
£  95  12s.  6d.,  exchange  at  4.87f . 

17.  How  large  a  draft  on  London  can  be  bought  for 
$646.29|,  exchange  at  4.88  ? 

18.  If  exchange  on  Germany  is  95^,  what  will  be  the 
face  of  a  draft  on  Hamburg  costing  f  358.37  ? 

19.  A  merchant  imported  2650  yards  of  silk,  invoiced  at 
7.20  francs  a  yard,  and  1200  yards  of  woollens,  invoiced  at 
6.70  francs  a  yard.  Find  the  cost  of  a  draft  on  Paris  for 
the  amount  of  the  bill,  exchange  at  5.16J. 


274  ARITHMETIC. 


XXII.    EQUATION  OF  PAYMENTS. 

349.  Equation  of  Payments  is  the  process  of  finding  at 
what  time  several  payments,  due  at  different  times,  may 
all  be  paid  at  once,  without  injustice  to  either  debtor  or 
creditor. 

The  time  thus  found  is  called  the  Equated  Time. 

EXAMPLES. 

350.  1.  A  owes  B  $150,  of  which  $25  is  due  in  3 
months,  $  50  in  4  months,  $  35  in  5  months,  and  $  40  in 
7  months.     What  is  the  equated  time  of  payment  ? 

25x3=  75 
50x4  =  200 
35  X  5  =  175 
40  X  7  =  280 
150  )  730(411  mo.  =  4  mo.  26  d.,  Ans. 

600 

130 

A  is  entitled  to  3  months'  use  of  $  25,  or  25  x  3  months'  use  of  $  1 
"        "        "4        "  "    "   $50,  or  50x4        "         "     "  $1 

"        "        "  5        "  "    "   |35,  or  35  X  5        "         "     "  $1 

u        u        u7        u  u    "$40,  or  40x7        "         "     "  $1. 

In  all,  A  is  entitled  to  730  months'  use  of  $  1. 

Then  he  is  entitled  to  the  use  of  $  150  for  as  many  months  as  150 
is  contained  times  in  730,  which  is  4if  months,  or  4  mo.  26  d. 

2.  What  is  the  average  time  of  paying  $  200  due  May  4, 
^300  due  June  12,  and  $400  due  July  24  ? 

200  X     0  =           0  We  select  the  earliest  date, 

300  X  39  =  11700  May  4,  as  a  convenient  date 

400  X  81  =  32400  from  which  to  reckon  times. 

900               )  44100  From  May  4  to  June  12  is 

Aq  J  39  d. ,  and  from  May  4  to  July 

May  4  +  49  d.  =  June  22,  Ans.  -p,.^«„^!q:^„  „„  ,-„  -p^  i   „,« 

•^  '  Proceeding  as  m  iiX.  1,  we 

find  the  equated  time  to  be  49  d.  after  May  4,  or  June  22. 


EQUATION  OF   PAYMENTS  275 

Note  1.  The  date  from  which  times  are  reckoned  is  called  the 
Focal  Date. 

Note  2.  If  there  is  a  common  term  of  credit,  we  may  find  the 
average  time  without  regard  to  that  term,  and  add  it  to  the  result. 

Thus,  if  goods  are  bought  on  60  days'  credit  as  follows  :  July  5, 
$  600 ;  Aug.  15,  $  400  ;  Sept.  10,  $  500 ;  we  find  the  equated  time  of 
payment  without  regard  to  the  term  of  credit,  and  add  60  days  to  the 
result. 

Note  3.  If,  in  any  result,  the  fraction  of  a  day  is  |,  or  more  than 
^,  it  is  reckoned  as  one  day  ;  but  if  it  is  less  than  ^,  it  is  disregarded. 

3.  A  owes  $  250  due  in  9  months ;  if  he  pays  $  75  in  4 
months,  and  $55  in  8  months,  when  should  he  pay  the 
balance  ? 

75  >^  5  _  375  By  paying  $  75  5  months  before  it  is 

55  X  j^  _.    55  due,  and  $  55  1  month  before  it  is  due, 

A  loses  the  use  of  $  1  for  75  x  5  +  55  x  1, 

120) 430 (3  i^g.  mo.  or  430  months. 

360  Hence,  he  is  entitled  to  the  use  of  the 

balance,  $  120,  long  enough  after  it  be- 

'  comes  due  to  be  equivalent  to  430  months' 

3  mo.  18  d.,  Ans.  use  of  $1. 

Then  he  is  entitled  to  the  use  of  the  balance  for  as  many  months 
after  it  becomes  due  as  120  is  contained  times  in  430 ;  that  is,  S/^ 
months,  or  3  mo.  18  d. 

4.  What  is  the  average  time  of  paying  $55  due  in  3 
months,  $  170  due  in  9  months,  and  $  135  due  in  7  months  ? 

5.  A  owes  B  $300,  of  which  $45  is  due  in  6  months, 
$  85  in  8  months,  $  75  in  9  months,  and  $  95  in  11  months. 
What  is  the  equated  time  of  payment  ? 

6.  Find  the  equated  time  of  paying  $  420  due  in  30  days, 
$  720  due  in  60  days,  $  120  due  in  90  days,  and  $  540  due  in 
120  days. 

7.  What  is  the  equated  time  of  paying  $  15  due  March 
15,  $  135  due  April  6,  and  $90  due  May  25  ? 

8.  Find  the  average  time  of  paying  $  175  due  Aug.  29, 
$340  due  Sept.  23,  $225  due  Oct.  5,  and  $410  due  Nov.  13. 


276  ARITHMETIC. 

9.  On  May  23,  I  bought  a  piece  of  land  for  $  1300,  on 
four  months'  credit.  If  I  pay  $625  on  July  23,  when 
should  I  pay  the  balance  ? 

10.  If  goods  are  bought  on  three  months'  credit  as  fol- 
lows :  Sept.  19,  1889,  $  1150 ;  Oct.  3,  1889,  $  925 ;  Nov.  12, 
1889,  $  775  ;  what  is  the  equated  time  of  payment  ? 

11.  Four  sixty -day  notes  bear  date  as  follows :  Jan.  4, 

1891,  f  565;  Feb.  27,  1891,   $350;  June  18,  1891,  $495; 
July  30, 1891,  $  210.    What  is  the  average  date  of  payment  ? 

Note.     A  sixty-day  note  falls  due  63  days  after  its  date. 

12.  On  Oct.  21,  Henry  Williams  owed  $  157.25  due  in  40 
days,  $223.75  due  in  60  days,  and  $186  due  in  90  days. 
What  is  the  equated  date  of  payment  ? 

13.  A  bill  of  $1200  is  due  in  5  months  from  Jan.  13, 

1892.  If  payments  are  made  as  follows  :  $  259,  March  20, 
1892 ;  $  248,  May  5,  1892 ;  when  is  the  balance  due  ? 

14.  A  tradesman  owes  $  300  due  in  5  months,  and  $  750 
due  in  9  months ;  if  at  the  end  of  5  mo.  20  d.  he  pays  $  450, 
wl)en  should  the  balance  be  paid  ? 

15.  Hooker  and  Ingalls  bought  merchandise  on  60  days' 
credit  as  follows  :  June  28,  $  72.30 ;  Aug.  1,  $  156.75 ;  Sept. 
4,  $  95.10.     What  is  the  average  date  of  payment  ? 

16.  A  merchant  owes  $2350  due  in  10  months.  If  he 
pays  $400  in  2  months,  $350  in  4  months,  and  $525  in  8 
months,  when  should  he  pay  the  balance  ? 

17.  Four  ninety-day  notes  bear  date  as  follows :  March 
9,  1892,  $388.05;  May  24,  1892,  $254.75;  Aug.  13,  1892, 
$525;  Oct.  30,  1892,  $409.20.  What  is  the  average  date 
of  payment  ? 

AVERAGE  OF  ACCOUNTS. 

351.  Average  of  Accounts  is  the  process  of  finding  at 
what  time  the  balance  of  an  account  may  be  paid,  without 
injustice  to  either  debtor  or  creditor. 


EQUATION  OF  PAYMENTS. 


277 


EXAMPLES. 

352.   1.   Find  the  equated  time  for  paying  the  balance  of 
the  following  account : 


Dr. 

Charles 

Stuart. 

Cr. 

1892. 

1892. 

May  28 

To  Mdse.  30  d. 

$350 

June    8 

By  Draft,  30  d. 

$275 

June  16 

((           u 

125 

July  15 

"  Cash, 

250 

July     7 

"      "    2  mo. 

275 

Aug.  21 

U           (( 

125 

Solution. 

June  21, 

350x11=    3850 

July  11, 

275  X  25  = 

6875 

June  16, 

125x0    =         0 

July  15, 

250  X  29  = 

7250 

Sept.    1, 

275  X  83  =  22825 

Aug.  21, 

125  X  66  = 

8250 

750 
650 


26675 
22375 


650 


^2375 


$100bal.         4300  bal. 
4300  -  100  =  43  d. 
June  16  +  43  d.  =  July  29,  Ans. 

The  sum  of  the  items  on  the  debit  side  of  the  account  is 
^  750,  and  on  the  credit  side  f  650. 

Subtracting  $  650  from  ^  750,  there  is  a  balance  of  $  100 
on  the  debit  side  of  the  account,  showing  the  amount  still 
due  from  Charles  Stuart. 

Payment  for  the  merchandise  sold  May  28  is  due  30  days 
after  May  28,  or  June  27  ;  and  payment  for  that  sold  July  7 
is  due  2  months  after  July  7,  or  Sept.  7. 

The  30-day  draft  dated  June  8  is  due  33  days  after  June  8, 
or  July  11. 

Thus,  June  16  is  the  earliest  date  at  which  any  item 
becomes  due. 

Using  June  16  as  the  focal  date,  we  find  that  the  sum  of 
the  products  on  the  debit  side  of  the  account  is  equivalent 
to  the  use  of  $  1  for  26675  days. 


278  ARITHMETIC. 

Also,  the  sum  of  the  products  on  the  credit  side  is  equiva- 
lent to  the  use  of  $  1  for  22375  days. 

Subtracting  22375  from  26675,  there  is  a  balance  on  the 
debit  side  of  the  account,  equivalent  to  the  use  of  $  1  for 
4300  days. 

Now  in  order  to  make  the  sum  of  the  products  on  the 
credit  side  equal  to  the  sum  of  the  products  on  the  debit 
side,  it  is  evident  that  the  balance  of  $  100  must  be  paid  at 
some  date  after  the  focal  date. 

That  is,  Mr.  Stuart  can  settle  the  account  equitably  by 
paying  $  100  as  many  days  after  June  16  as  is  equivalent 
to  the  use  of  $1  for  4300  days. 

But  the  use  of  $  1  for  4300  days  is  equivalent  to  the  use 
of  $100  for  as  many  days  as  100  is  contained  times  in 
4300;  that  is,  43  days. 

Hence,  he  can  settle  the  account  equitably  by  paying 
f  100  43  days  after  June  16 ;  that  is,  on  July  29. 

If,  in  the  above  example,  the  sum  of  the  products  on  the 
a-edit  side  of  the  account  had  been  greater  by  4300  than 
the  sum  of  the  products  on  the  debit  side,  an  earlier  focal 
date  could  have  been  found,  which  would  have  made  the 
sum  of  the  products  on  the  credit  side  equal  to  the  sum  of 
the  products  on  the  debit  side. 

If,  for  example,  the  focal  date  had  been  43  days  earlier 
than  June  16,  the  sum  of  the  products  on  the  debit  side 
would  have  been  greater  by  750  x  43,  while  the  sum  of  the 
products  on  the  credit  side  would  have  been  greater  by 
650  X  43 ;  and  since  750  x  43  exceeds  650  x  43  by  4300,  this 
would  have  made  the  sum  of  the  products  on  the  credit  side 
equal  to  the  sum  of  the  products  on  the  debit  side. 

This  earlier  focal  date  would  then  have  been  the  equated 
time  of  payment. 

It  is  evident  from  the  above  that  if  the  two  balances  are 
on  the  same  side  of  the  account,  the  equated  time  is  after 
the  focal  date ;  but  if  they  are  on  opposite  sides  of  the 
account,  the  equated  time  is  before  the  focal  date. 


EQUATION  OF   PAYMENTS.  279 

From  the  above  example  we  derive  the  following 

RULE. 

Write  to  the  left  of  each  item  of  the  account  its  date  of 
maturity ;  and  select  as  a  focal  date  the  earliest  date  at  which 
any  item  becomes  due. 

Multiply  each  item  by  the  number  of  days  betiveen  its  date 
of  maturity  and  the  focal  date,  and  add  the  products  on  each 
side  of  the  account. 

Divide  the  balance  of  the  sums  of  the  products  by  the  balance 
of  the  account,  giving  the  number  of  days  between  the  focal 
date  and  the  equated  time  of  payment. 

If  the  balaiices  are  on  the  same  side  of  the  account,  the 
equated  time  is  after  the  focal  date;  if  they  are  on  opposite 
sides,  the  equated  time  is  before  the  focal  date. 

2.  Find  the  equated  time  for  paying  the  balance  of  the 
following  account : 


Dr. 

George  Adams. 

Cr. 

1891. 

1891. 

Sept.  16 

To  Mdse.  30  d. 

$550 

Oct.     4 

By  Cash, 

$500 

Oct.  28 

U              (( 

375 

Nov.    9 

u         u 

325 

3.   Find  the  equated  time  for  paying  the  balance  of  the 
following  account : 


Dr. 

Henry  Cole. 

Cr. 

1890. 

1890. 

Nov.  12 

To  Mdse.  30  d. 

$620 

Dec.  24 

By  Cash, 

$840 

Dec.    8 

"  •     "2  mo. 

475 

1891. 

1891. 

Feb.    7 

"  Mdse. 

760 

Jan.  25 

u          u 

745 

280 


ARITHMETIC. 


4.   Find  the  equated  time  for  paying  the  balance  of  the 
following  account : 

Dr.  William  Blake.  Cr. 


1892. 

1892. 

Jan.  18 

To  Mdse.  1  mo. 

^450 

Jan.  31 

By  Draft,  30  d. 

$300 

Feb.    6 

»      "      60  d. 

600 

April  4 

"  Cash, 

550 

5.   At  what  date  should  the  balance  of  the  following 
account  begin  to  draw  interest  ? 

Dr.  Edward  Dodge.  Cr. 


1892. 

1892. 

May    5 

To  Mdse.  1  mo. 

fl65 

May  18 

By  Cash, 

$115 

June  20 

u         u 

280 

June  29 

((          u 

150 

July  11 

"       "      60  d. 

105 

July  10 

"  Draft,  60  d. 

175 

6.    Find  the  equated  time  for  paying  the  balance  of  the 
following  account : 

Dr.  Richard  Hayes.  Cr. 


1891. 

1891. 

Feb.  23 

To  Mdse.  30  d. 

i$275 

Mar.  19 

By  Draft, 3  mo. 

$200 

Mar.  15 

U              (( 

420 

April  6 

"  Canh, 

175 

May    9 

"       "     2  mo. 

355 

May  31 

"  Draft,  30  d. 

450 

7.   Find  the  equated  time  for  paying  the  balance  of  the 
following  account : 

Dr,  John  Evans.  Cr. 


1892. 

1892. 

Aug.  23 

To  Mdse.  90  d. 

$345 

Sept.   2 

By  Draft,  90  d. 

$275 

Sept.  16 

"       "     2  mo. 

775 

Nov.  18 

"  Cash, 

450 

Nov.  13 

"       "     3mo. 

530 

Dec.     7 

"  Draft,  2  mo. 

325 

Dec.  28 

"  Cash, 

210 

EQUATION  OF  PAYMENTS. 


281 


8.    When  should  the  balance  of  the  following  account 
begin  to  draw  interest  ? 

Dr.  James  French.  Cr. 


1890. 

1890. 

Dec.  21 

ToMdse.eOd. 

$380 

Dec.  19 

By  Draft,  2  mo. 

$350 

1891. 

1891. 

Jan.  15 

u         u 

560 

Jan.     3 

"  Mdse. 

1045 

Feb.  27 

"     2  mo. 

420 

Mar.  16 

"  Draft,  60  d. 

1000 

April  1 

"       "     30  d. 

700 

Note  1.  The  latest  date  at  which  any  item  becomes  due  may  be 
taken  as  a  focal  date  ;  in  such  a  case,  if  the  balances  are  on  the  same 
side  of  the  account,  the  equated  time  is  before  the  focal  date  ;  if  they 
are  on  opposite  sides,  the  equated  time  is  after  the  focal  date. 

Note  2.  In  settling  an  account  in  which  the  sum  of  the  items  on 
the  debit  side  equals  the  sum  of  the  items  on  the  credit  side,  it  may 
happen  that  the  sum  of  the  products  on  one  side  of  the  account  is 
greater  than  the  sum  of  the  products  on  the  other  side. 

Thus,'  in  settling  the  following  account : 

Daniel  Green. 


Dr. 


Cr. 


1892. 

1892. 

May    5 

To  Mdse. 

$130 

May  17 

By  Cash, 

$150 

June  12 

(4             U 

210 

June  25 

it      ii 

190 

with  May  5  as  the  focal  date,  we  find : 

130  X    0  =       0 
•     210  X  38  =  7980 
$340  7980 

340 
Obal. 


150  X  12  =    1800 

190  X  51  =    9690 

$340 


11490 
7980 
3510  bal. 


It  appears  from  the  above  that  there  is  still  due  from  Mr.  Green  the 
use,  ov  interest,  of  $1  for  3510  days  ;  which,  at  6%,  is  $0.59. 

If  the  balance  of  3510  had  been  on  the  debit  side  of  the  account, 
settlement  would  be  made  by  paying  Mr.  Green  $  0.59. 


282  ARITHMETIC. 


THE  INTEREST  METHOD. 


353.   Another  method  of  averaging  accounts  is  known  as 
tiie  Interest  Method. 

"  We  will  select  for  illustration  the  example  which  is  solved 
by  the  product  method  on  page  277. 

We  will  suppose  that  the  account  is  settled  on  Sept.  7, 
the  latest  date  at  which  any  item  becomes  due. 


Solution. 

Int.  of  $350  for  72  d.  =  $4.20 
"     "      125    ''   83  d.  =     1.7292 
"     "     275   "     Od.  =        0 

$750                      $5.9292 
650                         5.2625 
$100bal.              $0.6667  ba 

Int. 

u 

1. 

of  $  275  for  58  d. 
"     250    "   54  d. 
"      125    "   17  d. 
$660 

=  $2.6583 
=     2.25 
=       .3542 
$  5.2625 

Int.  of  $  100  for  1  d.  =  $.0167. 

.6667  -^  .0167  =  40  d. 

Sept.  7  -  40  d.  =  July  29,  Ans. 

The  balance  of  the  account  is  $  100  on  the  debit  side. 

Payment  for  the  merchandise  bought  May  28  is  due  June 
.27 ;  if  it  is  not  made  until  Sept.  7,  Mr.  Stuart  should  pay 
interest  on  the  amount  for  72  days;  which,  at  6%,  is 
14.20. 

In  like  manner,  he  should  pay  interest  on  $  125  for  83 
days,  which  is  $  1.7292. 

Hence,  if  the  account  is  settled  Sept.  7,  he  should  pay 
$  5.9292  interest  in  addition  to  the  sum  of  the  items  on  the 
debit  side. 

Now  on  June  8  he  gave  his  draft,  at  30  days,  for  $  275, 
which  became  due  July  11 ;  in  paying  this  sum  58  days 
before  Sept.  7,  he  is  entitled  to  interest  on  the  amount  for 
58  days,  which  is  $  2.6583. 

In  like  manner,  he  is  entitled  to  interest  on  $  250  for  54 
days,  which  is  $  2.25,  and  on  $  125  for  17  days,  which  is 
$0.3542. 


EQUATION   OF   PAYMENTS.  283 

Hence,  if  the  account  is  settled  Sept.  7,  he  is  entitled  to 
interest  to  the  amount  of  $  5.2625. 

This  leaves  a  balance  of  interest  due  from  him  of  $  5.9292 
—  $  5.2625,  or  $  .6667 ;  that  is,  if  he  settles  the  account 
Sept.  7,  he  must  pay  $  .6667  in  addition  to  the  balance  of 
the  account. 

But  by  paying  the  $  100  a  little  earlier  than  Sept.  7,  he 
can  offset  the  interest  charge ;  and  the  question  now  is, 
how  many  days  will  it  take  $  100  to  gain  $  .6667  at  6%  ? 

In  one  day,  $  100  gains  $  .0167. 

Then,  to  gain  $  .6667  will  take  as  many  days  as  .0167  is 
contained  times  in  .6667  ;  which  is  40. 

Then,  by  paying  the  $  100  40  days  before  Sept.  7,  that  is, 
on  July  29,  he  can  settle  the  account  with  equity. 

It  is  evident  that  if  the  balances  had  been  on  opposite 
sides  of  the  account,  in  the  above  example,  the  equated  time 
of  payment  would  have  been  after  Sept.  7. 

From  the  above  example,  we  derive  the  following 

RULE. 

Select  as  a  focal  date  the  latest  date  at  ivhich  any  item 
becomes  due. 

Compute  interest  at  Q^o  on  eacJi  item  for  the  number  of  days 
from  the  date  when  it  becomes  due  to  the  focal  date. 

Divide  the  balance  of  interest  by  the  interest  of  the  balance 
of  the  account  for  one  day,  giving  the  number  of  days  from  the 
focal  date  to  the  equated  time  of  payment. 

If  the  balances  are  on  the  same  side  of  the  account,  the 
equated  time  is  earlier  than  the  focal  date;  if  they  are  on 
opposite  sides,  it  is  later  than  the  focal  date. 

The  teacher  may  have  the  examples  of  Art.  352  performed 
by  the  interest  method. 


284  ARITHMETIC. 


XXIII.    STOCKS    AND    BONDS. 

354.  Stock  is  the  capital  of  a  corporation ;  it  is  divided 
into  a  certain  number  of  equal  parts  called  Shares. 

355.  The  original  value  of  a  share  is  usually  $  100 ;  it 
will  be  so  considered  in  the  present  chapter,  unless  the  con- 
trary is  stated. 

356.  The  original  value  of  a  share  of  stock  is  called  its 
Par  Value,  and  the  price  at  which  it  sells  is  called  its 
Market  Value. 

357.  If  a  share  of  stock  sells  for  more  than  its  par  value, 
the  stock  is  said  to  be  above  par,  or  at  a  premium;  if  it 
sells  for  less  than  its  par  value,  the  stock  is  said  to  be  below 
par,  or  at  a  discount. 

358.  The  market  value  of  a  stock  is  usually  quoted  at  a 
certain  per  cent  of  the  par  value. 

Thus,  if  a  stock  is  quoted  at  105,  it  is  selling  for  5% 
above  its  par  value ;  if  it  is  quoted  at  95,  it  is  selling  for 
5%  below  its  par  value. 

359.  A  Certificate  of  Stock  is  a  document  issued  by  a 
corporation,  specifying  the  number  of  shares  owned  by  the 
holder,  and  the  par  value  of  each  share. 

A  Dividend  is  a  sum  divided  among  the  stockholders  from 
the  profits  of  the  business. 

An  Assessment  is  a  sum  required  of  the  stockholders  to 
meet  the  losses  or  expenses  of  the  business. 

Dividends  and  Assessments  are  generally  reckoned  at  a 
certain  per  cent  of  the  par  value  of  the  stock. 

360.  A  Bond  is  the  interest-bearing  note  of  a  government 
or  corporation. 

The  interest  on  bonds  is  usually  paid  semi-annually. 
A  Coupon  is  a  certificate  of  interest  attached  to  a  bond. 


STOCKS   AND  BONDS.  285 

361.  Bonds  are  usually  named  according  to  their  rate  of 
interest  and  date  of  maturity. 

Thus,  "  U.  S.  4i-'s.  '91 "  signifies  Bonds  issued  by  the 
United  States  governmentj  bearing  4J%  interest,  the  prin- 
cipal payable  in  1891. 

362.  Brokerage  is  the  commission  received  by  a  broker 
for  buying  or  selling  stocks  and  bonds. 

It  is  usually  |^%  of  the  par  value  of  the  stock  or  bond. 

Note.  It  will  be  understood,  in  the  following  examples,  that  the 
brokerage  is  not  included  in  the  quoted  price  of  a  stock. 

Thus,  if  a  man  buys  stock  at  112  J,  and  the  brokerage  is  ^%,  he 
pays  the  broker  112| ;  if  he  sells  stock  at  112^,  and  the  brokerage  is 
J%,  he  receives  only  1 12 1  from  the  broker. 

EXAMPLES. 

363.  1.   Find  the  cost  of  20  shares  New  York  Central 

stock,  at  112J,  brokerage  ^%. 

Since  the  cost  of  one  share, 

1121+    |-=112|-.  including  brokerage,    is   $112^, 

$  112|-  X  20  =  $  2257.50,  Ans.     the  cost  of   20  shares  will  be 

20  X  $112|,  or  $2257.50. 

2.  How  much  will  be  received  from  the  sale  of  16  shares 
Chicago,  Burlington,  and  Quincy  stock  at  103|,  brokerage 

Since  the  price  received  for  one 
103|  —  I"  =  103|-.  share    is  $  103^,  the  price  received 

$  103^  X  16  =  $  1656,  Ans.     for  16  shares  will  be  16  x  $  103^,  or 

$  1656, 

3.  What  amount  of  Mexican  Central  4's,  at  64i,  can  be 
bought  for  $16062.50,  including  a  brokerage  of  i%  ? 

641-  -I-  1-  =  641  =  64.25.  Since  the  cost  of  one  dollar's 

16062.5 -.6425  =  125000,  .In..     Zf.ot"'"'^'"^ ^\f""^^^"'' It 

'  $  0. 6425,  as  many  dollars'  worth 

can  be  bought  for  1 16062.50  as  .6425  is  contained  times  in  16062.5. 

Dividing  16062.5  by  .6425,  the  quotient  is  25000. 

Hence,  $25000  worth  of  bonds  can  be  bought  for  $  16062.50. 


286  ARITHMETIC. 

4.  If  48  shares  Union  Pacific  stock  are  sold  for  $  1992, 
brokerage  J%,  what  is  the  quoted  price  of  the  stock  ? 

$  1992  ^  48  =  $  41.50  =  1 41f  Dividing   $  1992  by  48,  the 

4i  I    ,    ^        I  -1 Q      A  price  received  for  one  share  is 

Then  the  quoted  price  is  41^,  plus  the  brokerage  of  I,  or  41|. 

5.  Find  the  cost'  of  95  shares  Old  Colony  stock,  at  186f, 
brokerage  ^%. 

6.  Find  the  cost  of  84  shares  of  telegraph  stock  at  95J, 
brokerage  J%. 

7.  How  much  will  be  received  from  the  sale  of  37  shares 
of  bank  stock  at  147f,  brokerage  i-%  ? 

8.  A  man  sold  5S  shares  of  New  York  and  New  England 
stock  at  44|,  brokerage  i%.     How  much  did  he  receive  ? 

9.  Find  the  cost  of  $  12000  U.  S.  4's,  when  at  a  premium 
of  12|%,  brokerage  i%. 

10.  Find  the  cost  of  72  shares  of  railway  stock,  when 
31|-%  below  par,  brokerage  J%. 

11.  How  many  shares  of  Missouri  Pacific  stock,  at  56|-, 
can  be  bought  for  $2604.75,  including  a  brokerage  of  |^%  ? 

12.  A  gentleman  sold  bonds  at  88|,  brokerage  i%,  re- 
ceiving the  sum  of  $  6637.50.  What  amount  of  bonds  did 
he  sell  ? 

13.  I  sold  mining  stock  at  126J,  brokerage  ^%,  receiving 
the  sum  of  $  5450.25.     How  many  shares  did  I  sell  ? 

14.  How  many  shares  of  railway  stock,  at  $151.75  a 
share,  can  be  bought  for  $95073.75,  including  a  brokerage 
of  i%  ? 

15.  What  amount  of  Missouri,  Kansas,  and  Texas  5's,  at 
a  discount  of  51|%,  can  be  bought  for  $7237.50,  including 
a  brokerage  of  |%  ? 

16.  How  many  shares  of  bank  stock,  at  27|%  above  par, 
can  be  bought  for  $  9581.25,  including  a  brokerage  of  J%  ? 


STOCKS '^^tt  BONDS.  287 

17.  If  12  shares  of  Elevated  Eailway  stock  are  sold  for 
f  1837.50,  including  a  brokerage  of  i%,  what  is  the  quoted 
price  of  the  stock  ? 

18.  If  45  shares  Boston  and  Maine  stock  can  be  bought 
for  $7627.50,  including  a  brokerage  of  i%,  what  is  the 
quoted  price  of  the  stock  ? 

19.  If  $6000  worth  of  bonds  are  sold  for  $5497.50, 
including  a  brokerage  of  ^%,  how  much  below  par  are  the 
bonds  quoted  ? 

20.  If  18  shares  of  railway  stock  can  be  bought  for 
$1460.25,  including  a  brokerage  of  ^%,  what  is  the  quoted 
price  of  the  stock  ? 

21.  If  $7500  Iowa  Central  bonds  can  be  bought  for 
$6581.25,  including  a  brokerage  of  i%,  at  what  per  cent 
discount  are  the  bonds  quoted  ? 

22.  If  298  shares  Delaware  and  Hudson  stock  can  be 
bought  for  $40788.75,  including  a  brokerage  of  ^%,  how 
much  above  par  is  the  stock  quoted  ? 

23.  If  the  brokerage,  at  ^%,  for  selling  stock  is  $23.25, 
how  many  shares  were  sold  ? 

24.  What  annual  income  is  received  from  mining  stocks 
whose  par  value  is  $11300,  paying  lf%  dividends  semi- 
annually ? 

25.  A  corporation  declared  a  dividend  of  7J%,  paying  to 
the  stockholders  a  total  amount  of  $  5437.50.  What  was  its 
capital  stock  ? 

26.  A  man  purchased  358  shares  of  a  certain  stock  at 
3|%  below  par,  and  sold  it  at  a  premium  of  7f  %  ;  if  he  paid 
^%  brokerage  on  each  transaction,  how  much  did  he  gain  ? 

27.  A  corporation  whose  capital  is  $225000,  levies  an 
assessment  of  $  4218.75  on  its  stockholders.  What  is  the 
rate  per  cent  of  the  assessment  ? 

28.  What  par  value  of  stocks  paying  1|%  dividends 
quarterly,  will  produce  an  annual  income  of  $  513.50  ? 


288  .  ARITHMETIC. 

29.  If  $  441.75  is  lost  by  buying  stocks  at  5J%  premium, 
and  selling  them  at  6J^%  below  par,  in  each  case  paying  a 
brokerage  of  -J-^,  how  many  shares  were  bought  ? 

30.  A  man  sold  204  shares  of  stock  at  80,  and  with  the 
proceeds  bought  stock  at  106|.  If  the  brokerage  on  each 
transaction  was  i%,  how  many  shares  did  he  buy  ? 

31.  A  man  sold  285  shares  of  railway  stock  at  103 J,  and 
invested  the  proceeds  in  bank  stock  at  78J.  If  the  broker- 
age on  each  transaction  was  ^%,  how  many  shares  of  bank 
stock  did  he  receive  ? 

32.  What  annual  income  will  be  realized  from  investing 
^2145  in  a  5%  stock  at  1071,  brokerage  -|-%  ? 

The  cost  of  the  stock,  in- 

iO'^i  +  i  =  ^•^'^i  =  107.25.  eluding  brokerage,  is  107.25.  ' 

$  2145  H-  1.0725  =  $  2000.  Then  stock  to  the  par  value 

$2000  X  .05  =  $  100,  A71S.      of  $2145-  1.0725,  or  $2000, 

can  be  bought  for  $2145. 
The  annual  income  from  $2000  at  5%  is  $2000  x  .05,  or  $  100. 

33.  What  amount  must  be  invested  in  a  5i%  stock,  at 
93f,  no  allowance  being  made  for  brokerage,  to  realize  an 
annual  income  of  $  374  ? 

^  374  -^  .055  =  $  6800.  K  the  annual  income  is  to  be 

<Di^oAAw    OQ^       as^aQry;^     a^„      $ 374,  the  par  valuc  of  the  stock 
^  bmO  X  .y^^  =  *  bd7o,  Ans.     ^^^^  ^^  ^  3^^  ^  _^^^^  ^^  ^  gg^^_ 

To  purchase  $  6800  worth  of  stock  at  93|,  the  amount  to  be  invested 
is  $6800  X  .93|,  or  $6375. 

34.  If  I  purchase  at  95f,  brokerage  J%,  a  stock  paying 
6%  dividends  annually,  what  per  cent  does  the  investment 
yield  ? 


95|.  +  4  =  96.  The  cost  of  the  stock,  including  broker- 

_6_  =  JL  =  6i^.    Ans      age,  is  $96  a  share. 

9  6        16  ¥/t^'  •         Then,  if  $  96  produces  an  annual  income 

of  $6,  the  rate  per  cent  of  the  investment  is  -q%,  or  6|%. 


STOCKS   AND  BONDS.  289 

35.  What  must  be  paid  for  a  5|%  stock,  no  allowance 
being  made  for  brokerage,  in  order  that  the  investment  may 
yield  5%  ? 

^_K        ^^.       ^^K     A  The  annual  income  from  one  share  is 

5.75  -r-  .05  =  115,  Ans.     ^^  -  -^ 

Then  if  $5. 75  is  5%  of  the  cost  of  a  share,  the  cost  of  a  share  must 
be  $5.75 -.05,  or  $115. 

Hence,  the  stock  must  he  bought  at  115. 

36.  What  annual  income  will  be  realized  from  investing 
$3455  in  4%  bonds  at  86f,  no  allowance  being  made  for 
brokerage  ? 

37.  What  annual  income  will  be  realized  from  investing 
$2805  in  a  6J%  stock  at  116|,  brokerage  \%  ? 

38.  What  annual  income  will  be  realized  by  investing 
$4331.25  in  a  7f  %  stock,  at  a  premium  of  23^%,  brokerage 

39.  What  annual  income  will  be  realized  by  investing 
$  7879.50  in  a  4|%  stock,  at  22f  %  below  par,  no  allowance 
being  made  for  brokerage  ? 

40.  What  sum  must  be  invested  in  a  5J%  stock  at  105|^, 
no  allowance  being  made  for  brokerage,  to  realize  an  annual 
income  of  $220.50? 

41.  What  sum  must  be  invested  in  3%  bonds  at  96^, 
brokerage  |%,  to  yield  an  annual  income  of  $  180  ? 

42.  What  sum  must  be  invested  in  a  4|-%  stock  at  a  dis- 
count of  16J%,  brokerage  J%,  to  yield  an  annual  income  of 

$648? 

43.  A  man  realized  an  annual  income  of  $425.25  by 
investing  in  a  6f  %  stock,  at  18  J  %  above  par,  no  allowance 
being  made  for  brokerage.     What  was  the  sum  invested  ? 

44.  If  a  5J%  stock  is  purchased  at  105,  no  allowance 
being  made  for  brokerage,  what  per  cent  does  the  invest- 
ment yield  ? 


290  ARITHMETIC. 

45.  If  a  7|%  stock  is  bought  at  174J,  brokerage  i%, 
what  per  cent  does  the  investment  yield  ? 

46.  If  3%  bonds  are  bought  at  46f,  brokerage  ^%,  what 
per  cent  does  the  investment  yield  ? 

47.  If  4|-%  stock  is  bought  at  123 J,  no  allowance  being 
made  for  brokerage,  what  per  cent  does  the  investment 
yield? 

48.  What  must  be  paid  for  a  7%  stock,  no  allowance 
being  made  for  brokerage,  in  order  that  the  investment 
may  yield  3^%  ? 

49.  What  must  be  paid  for  a  4|%  stock,  no  allowance 
being  made  for  brokerage,  in  order  that  the  investment 
may  yield  6i%  ? 

50.  What  must  be  paid  for  a  5|-%  stock,  no  allowance 
being  made  for  brokerage,  in  order  that  the  investment 
may  yield  4%  ? 

51.  At  what  price  must  3|%  bonds  be  quoted,  in  order 
that  a  person  investing  in  them,  brokerage  ^%,  may  realize 
5|-%  for  his  money? 

52.  At  what  price  must  a  7%  bond  be  quoted  to  yield 
the  same  per  cent  on  the  investment  as  a  5%  bond  at  107^? 

53.  Which  will  yield  the  greater  per  cent  on  the  invest- 
ment, a  6%  stock  at  125,  or  a  4i%  stock  at  90,  no  allowance 
being  made  for  brokerage  ? 

54.  A  man  sells  6%  stock  to  the  par  value  of  ^7400  at 
113|,  and  invests  the  proceeds  in  5%  stock  at  92 1-,  no 
allowance  being  made  for  brokerage.  Is  his  income  in- 
creased or  diminished,  and  how  much  ? 

55.  At  what  price  must  a  3i%  stock  be  bought,  to  yield 
the  same  per  cent  on  the  investment  as  a  4i%  stock  at 
21^%  below  par? 

56.  Which  is  the  better  investment,  a  7%  stock  at  143f, 
or  a  5J%  stock  at  112|,  brokerage  ^%  on  each  transaction? 


STOCKS  AND  BONDS.  291 

57.  A  man  sells  4^%  stock  to  the  par  value  of  $9400  at 
82^,  and  invests  the  proceeds  in  8|%  stock  at  163 1,  bro- 
kerage on  each  transaction  ^%.  Is  his  income  increased  or 
diminished,  and  how  much  ? 

58.  Which  will  yield  the  greater  income,  f  6757.50  in- 
Vested  in  6%  bonds  at  112f,  or  $6632.50  invested  in  5% 
bonds  at  94|-,  brokerage  in  each  case  J%  ? 

59.  A  man  sold  $12000  5-i%  state  bonds  at  llOf,  and 
invested  the  proceeds  in  city  bonds  at  84f,  brokerage  in 
each  case  J%.  If  his  income  was  increased  $33,  what  per 
cent  were  the  city  bonds  ? 

60.  Which  will  yield  the  greater  per  cent  on  the  invest- 
ment, a  5^%  stock  at  a  premium  of  3}%,  or  a  3|%  stock 
at  a  discount  of  28^%,  brokerage  in  each  case  i%  ? 

61.  A  man  sold  51%  stock  to  the  par  value  of  $8600  at 
1041  and  invested  the  proceeds  in  6^%  stock,  decreasing 
his  income  by  $21.25.  If  no  allowance  be  made  for  bro- 
kerage, what  price  did  he  pay  for  the  6^%  stock  ? 

62.  A  man  sold  78  shares  of  5|- %  stock  at  1311,  and  43 
shares  of  3J%  stock  at  81|^,  and  invested  the  proceeds  in 
4f  %  stock  at  91f ;  brokerage  on  each  transaction  i%.  Was 
his  income  increased  or  diminished,  and  how  much  ? 


292  ARITHMETIC. 

XXIV.    PROGRESSIONS. 

ARITHMETICAL  PROGRESSION. 

364.  An  Arithmetical  Progression  is  a  series  of  numbers 
which  increase  or  decrease  by  a  constant  difference,  called 
the  Common  Difference. 

Thus,  1,  3,  5,  7,  9,  11  is  an  increasing  arithmetical  pro- 
gression, in  which  the  common  difference  is  2. 

Again,  14,  11,  8,  5,  2,  is  a  decreasing  arithmetical  pro- 
gression, in  which  the  common  difference  is  3. 

The  numbers  which  compose  the  progression  are  called 
its  Terms. 

365.  To  find  any  Term  of  an  Arithmetical  Progression. 

Example.  Find  the  18th  term  of  the  arithmetical  pro- 
gression 3,  9,  15,  etc. 

Com.  dif.   =9  —  3  =  6.  In  this  case,  the  common  differ- 

18th  term  =  3  +  (17  X  6)     ^^^e  is  6. 

=  3  _!_  102  ^oyf  the  second  term  is  equal  to 

_  .  the  first  term  plus  once  the  commou 

=  lUo,  Ans.  difference  ;  the  third  term  is  equal  to 

the  first  term  plus  twice  the  common  difference  ;  etc. 

Hence,  the  eighteenth  term  will  be  equal  to  the  first  term  plus 

seventeen  times  the  common  difference ;  that  is,  3  +  (17  x  6),  or  105. 

If  the  progression  had  been  a  decreasing  one,  we  should 
have  subtracted  seventeen  times  the  common  difference  from 
the  first  term. 

From  the  above  example,  we  derive  the  following 

RULE. 

To  find  any  term  of  an  arithmetical  progression,  add  to  or 
subtract  from  the  first  term,  according  as  the  progression  is 
increasing  or  decreasing,  the  common  difference  multiplied  by 
a  number  less  by  1  than  the  number  of  the  required  term. 


rROGKESSIONS.  293 

366.  To  find  the  Sum  of  the  Terms  of  an  Arithmetical 
Progression. 

Example.  Find  the  sum  of  the  terms  of  the  arithmetical 
progression  5,  9,  13,  17,  21,  25,  29. 

5+9  +  13  +  17  +  21  +  25  +  29 
29  +  25  +  21  +  17  +  13+   9+   5 

34  +  34  +  34  +  34  +  34  +  34  +  34 

The  sum  of  the  terms  of  the  progression  is  5  +  9  +  1 3 +17  +  21  + 
25  +  29. 

We  write  underneath  this  the  sum  of  the  terms  of  the  progression 
in  reverse  order;  that  is,  29  +  25  +  21  +  17  +  13  +  9  +  5. 

Adding  each  term  of  the  second  line  to  the  term  directly  above  it, 
the  sum  is  34. 

Hence,  twice  the  sum  of  the  terms  is  7  x  34,  or  7  x  (5  +  29). 

Therefore  the  sum  of  the  terms  is  Z_><1^_±_29)    qj,  hq  ^^g 

2 

Observing  that  7  is  the  number  of  terms,  5  the  first  term, 
and  29  the  last  term,  we  have  the  following 

RULE. 

To  find  the  sum  of  the  terms  of  an  arithmetical  progression, 
multiply  the  sum  of  the  first  and  last  terms  by  the  number  of 
terms,  and  divide  the  result  by  2. 

EXAMPLES. 

367.  1.  Find  the  last  term  and  the  sum  of  the  terms  of 
the  arithmetical  progression  19,  IS^,  17|,  etc.,  to  24  terms. 

The  common  difference  is  19  —  18},  or  |, 

Since  the  number  of  terfhs  is  24,  the  last  term  is  the  24th  term. 

By  Art.  365,  the  24th  term  is  19  -  (23  x  f),  or  3|. 

By  Art.  366,  the  sum  of  the  terms  is  24  x  (19  +  3|)^  ^^  272. 

2.  Find  the  11th  term  of  the  progression  2,  9,  16,  etc. 

3.  Find  the  38th  term  of  the  progression  4|,  h\,  6^,  etc. 

4.  Find  the  23d  term  of  the  progression  327,  316,  3*05,  etc. 


294  ARITHMETIC. 

5.  Find  the  20th  term  of  the  progression  120,  116.4, 
112.8,  etc. 

6.  Find  the  47th  term  of  the  progression  -^q,  j\,  -^-^,  etc. 

Find  the  last  term  and  the  sum  of  the  terms  of : 

7.  4,  10,  16,  etc.,  to  12  terms. 

8.  9,  23,  37,  etc.,  to  21  terms. 

9.  293,  285,  277,  etc.,  to  29  terms. 

10.  486,  473,  460,  etc.,  to  36  terms. 

11.  3J,  5,  6J,  etc.,  to  48  terms. 

12.  97,  90.3,  83.6,  etc.,  to  14  terms. 

13.  2^  3^  5^  etc.,  to  59  terms. 

14.  I,  I,  If,  etc.,  to  23  terms. 

15.  Find  the  sum  of  the  integers  beginning  with  1,  and 
ending  with  99. 

16.  Find  the  sum  of  the  even  integers  beginning  with  2, 
and  ending  with  100. 

17.  Continue  the  progression  y^g,  ■^,  -J,  etc.,  to  four 
more  terms. 

18.  Find  the  sum  of  the  first  18  integers  which  are 
multiples  of  7. 

19.  Find  the  sum  of  all  the  multiples  of  11,  from  110  to 
990,  inclusive. 

20.  A  body  falls  16^2  ^^^^  *^®  ^^^  second,  and  in  each 
succeeding  second  32i-  feet  more  than  in  the  next  preceding 
one.  How  far  will  it  fall  in  the  16th  second  ?  How  far 
will  it  fall  in  16  seconds  ? 

21.  A  man  travelled  43^  miles  the  first  day,  and  on  each 
succeeding  day  2|  miles  less  than  on  the  next  preceding. 
How  £ar  did  he  travel  on  the  11th  day  ?  How  far  did  he 
travel  in  11  days  ? 


PROGRESSIONS.  295 

22.  If  a  person  saves  ^  100  a  year,  and  puts  this  sum  at 
simple  interest  at  4|^%  at  the  end  of  each  year,  to  how  much 
will  his  property  amount  at  the  end  of  25  years  ? 

GEOMETRICAL  PROGRESSION. 

368.  A  Geometrical  Progression  is  a  series  of  numbers 
which  increase  or  decrease  by  a  constant  multiplier,  called 
the  Ratio. 

Thus,  1,  3,  9,  27,  81  is  an  increasing  geometrical  progres- 
sion, in  which  the  ratio  is  3. 

Again,  64,  32,  16,  8,  4,  is  a  decreasing  geometrical  pro- 
gression, in  which  the  ratio  is  ^. 

The  numbers  which  compose  the  progression  are  called 
its  Terms. 

369.  To  find  any  Term  of  a  Geometrical  Progression. 

Example.  Find  the  6th  term  of  the  geometrical  progres- 
sion 2,  6,  18,  etc. 

Dividing  the  second  term,  6,  by  the  first 
Ratio  =  1  =  3.  term,  2,  the  ratio  is  3. 

6th  term  =  2x3^  Now  the  second  term  is  equal  to  the  first 

=  2  X  243        term  times  the  first  power  of  the  ratio  ;  the 
=  486   Ans      ^^*'^^  term  is  equal  to  the  first  term  times 
the  second  power  of  the  ratio  ;  etc. 
Hence,  the  sixth  term  will  be  equal  to  the  first  term  times  the  fifth 
power  of  the  ratio  ;  that  is,  2  x  3^,  or  486. 

From  the  above  example,  we  derive  the  following 

RULE. 

To  find  any  term  of  a  geometrical  progression,  multiply  the 
first  term  by  that  power  of  the  ratio  whose  exponent  is  less  by 
1  than  the  number  of  the  required  term. 

370.  To  find  the  Sum  of  the  Terms  of  a  Geometrical  Pro- 
gression. 

1.  Find  the  sum  of  the  terms  of  the  geometrical  pro- 
gression 2,  6,  18,  54,  162. 


296  ARITHMETIC. 

The  ratio  is  3. 

The  sum  of  the  terms  =2  +  6+18  +  54  + 162.  (I) 

Multiplying  each  term  of  the  result  by  the  ratio,  3,  we  have 

3  X  the  sum  of  the  terms      =  6  +  18  +  54  +  162  +  (162  x  3).        (2) 

Subtracting  (1)  from  (2),  we  have 

2  X  the  sum  of  the  terms      =  (162  x  3)  -  2. 

rr^u        .V.               *.!,    .               (162x3)- 2      (162x3)-2 
Then,  the  sum  of  the  terms  =  ^ ^ =  ^ — ^ — ^ 

Observing  that  162  is  the  last  term,  3  the  ratio,  and  2 
the  first  term,  we  have  the  following  rule : 

The  sum  of  the  terms  of  an  increashig  geometrical  progres- 
sion is  equal  to  the  product  of  the  last  term  by  the  ratio,  minus 
the  first  term,  divided  by  the  ratio  minus  1. 

2.  Find  the  sum  of  the  terms  of  the  geometrical  progres- 
sion 768,  192,  48,  12,  3. 


The  sum  of  the  terms  =  768  +  192  +  48  +  12  +  3.  (1) 

Multiplying  each  term  of  the  result  by  the  ratio,  ^,  we  have 

^  X  the  sum  of  the  terms      =  192  +  48  +  12  +  3  +  (3  x  \).         (2) 

Subtracting  (2)  from  (1),  we  have 

I  X  the  sum  of  the  terms      =  768  -  (3  x  I). 

768  -  (3  X  i)      768  -  (3  x  i) 


Then,  the  sum  of  the  terms  = 


f  1- 


Observing  that  768  is  the  first  term,'  3  the  last  term,  and 
i  the  ratio,  we  have  the  following  rule : 

The  sum  of  the  terms  of  a  decreasing  geometrical  progres- 
sion is  equal  to  the  first  term,  minus  the  product  of  the  last 
term  by  the  ratio,  divided  by  1  minus  the  ratio. 

371.   In  Ex.  1,  Art.  370,  we  have 

162  X  3  =  (2  X  30  X  3  =  2  X  3«. 

That  is,  the  product  of  the  last  term  of  a  geometrical 
progression  by  the  ratio  is  equal  to  the  first  term,  multi- 
plied by  that  power  of  the  ratio  whose  exponent  is  equal  to 
the  number  of  terms. 


PKOGRESSIONS.  297 

Then  the  rules  of  Art.  370  may  be  stated  as  follows : 

Tlie  sum  of  the  terms  of  an  increasing  geometrical  progres- 
sion is  equal  to  the  first  term,  multiplied  by  that  poiver  of 
the  ratio  whose  exponent  is  equal  to  the  number  of  terms,  minus 
the  first  term,  divided  by  the  ratio  minus  1. 

The  sum  of  the  terms  of  a  decreasing  geometi^ical  progres- 
sion is  equal  to  the  first  term,  minus  the  first  term  multiplied 
by  that  power  of  the  ratio  whose  exponent  is  equal  to  the  num- 
ber of  terms,  divided  by  1  minus  the  ratio. 

EXAMPLES. 

372.  1.  rind  the  last  term  and  the  sum  of  the  terms  of 
the  geometrical  progression  3,  12,  48,  etc.,  to -6  terms. 

The  ratio  is  -L2,  or  4. 

By  Art.  369,  the  last  or  6th  term  is  3x45,  or  3072. 

By  the  first  rule  of  Art.  370,  the  sum  of  the  terms  is 

4  X  3072  -  3   ^^  .^Q. 

,  or  4095. 

4-1       ' 

2.  Find  the  sum  of  the  terms  of  the  geometrical  pro- 
gression 2,  i,  f ,  etc.,  to  7  terms. 

The  ratio  is  |  ^  2,  or  i. 

By  the  second  rule  of  Art.  371,  the  sum  of  the  terms  is 

2 - 2  X ay_2- ^tW_ im_2186  ^^^ 

1  -  i        ~        I        ""     I     -  729  ' 

3.  Find  the  5th  term  of  the  progression  4,  20,  100,  etc. 

4.  Find  the  6th  term  of  the  progression  f ,  i,  |,  etc. 

5.  Find  the  9th  term  of  the  progression  6|,  41  3,  etc. 

6.  Find  the  7th  term  of  the  progression  600, 150,  37|,  etc. 

Find  the  last  term  and  the  sum  of  the  terms  of : 

7.  1,  2,  4,  etc.,  to  11  terms. 

8.  6,  18,  54,  etc.,  to  8  terms. . 

9.  20,  10,  5,  etc.,  to  10  terms. 


298  ARITHMETIC. 

10-   i  A»  H?  etc.,  to  6  terms. 

11.  -J-,  |-,  |,  etc.,  to  7  terms. 

12.  12|,  5,  2,  etc.,  to  5  terms. 

13.  Find  the  sum  of  the  terms  of  the  progression  12|,  16, 
20,  etc.,  to  -5  terms. 

14.  Find  the  sum  of  the  terms  of  the  progression  15,  10, 
6|,  etc.,  to  8  terms. 

15.  A  man  agreed  to  work  for  14  days  on  condition  that 
he  should  receive  1  cent  the  first  day,  2  cents  the  second 
day,  4  cents  the  third  day,  and  so  on.  How  much  did  he 
receive  in  all  ? 

16.  A  man  travelled  384  miles  the  first  day,  and  on  each 
succeeding  day  one-half  as  many  miles  as  on  the  next  pre- 
ceding. How  far  did  he  travel  on  the  10th  day  ?  How 
far  did  he  travel  in  10  days  ? 

17.  Continue  the  progression  4|f,  3|,  2|,  etc.,  to  three 
more  terms. 

18.  The  population  of  a  certain  city  at  the  end  of  each 
year  is  1.04  times  as  great  as  at  the  beginning  of  the  year. 
If  the  population  on  Jan.  1,  1890,  was  15625,  what  will  it 
be  on  Jan.  1,  1893  ? 

19.  If  the  first  term  is  $  100,  the  ratio  1.05,  and  the 
number  of  terms  5,  what  is  the  last  term  ? 

COMPOUND  INTEREST. 

373.  Problems  in  compound  interest  may  be  solved  by 
aid  of  the  principles  of  geometrical  progression. 

*  Thus,  let  $  100  be  put  at  compound  interest  at  6%. 
The  amount  at  the  end  of  one  year  is  $  100  x  1.06. 
The  amount  at  the  end  of  two  years  is  $  100  x  1.06  x  1.06, 
or  $100  X  (1.06)2;  and  so  on. 


PROGRESSIONS.  299 

Hence,  the  amount  at  the  end  of  any  number  of  years  is 
equal  to  the  principal,  multiplied  by  that  power  of  1  plus  the 
rate,  whose  expo7ient  is  equal  to  the  number  of  years, 

EXAMPLES. 

1.  Find  the  amount  of  $1600  for  4  years,  at  5%  com- 
pound interest. 

By  the  above  rule,  the  required  amount  is 

$1600  X  (1.05)*,  or  $1944.81,  Ans. 

2.  What  principal  will  gain  $15,608  in  3  years  at  4% 
compound  interest  ? 

The  amount  of  $1  for  3  years  at  4%  compound  interest  is  (1.04)* 
dollars,  or  $1.124864. 

Then,  $  1  will  gain  $  .124864  in  3  years  at  4%  compound  interest. 

Then,  to  gain  $15,608  will  take  as  many  dollars  as  .124864  is  con- 
tained times  in  15.608,  which  is  $  125,  Ans. 

3..  Find  tha^  amount  of  $6400  for  4  years,  at  3%  com- 
pound interest. 

4.  Find  the  amount  of  $8000  for  5  years,  at  4%  com- 
pound interest. 

5.  Find  the  compound  interest  of  $300  for  4  years, 
at^%. 

6.  Find  the  compound  interest  of  $760  for  3  years,  at 

7.  What  principal  will  amount  to  $  25585.35  in  2  years, 
at  3J%  compound  interest  ? 

8.  What  principal  will  amount  to  $5477.5974»in  3  years, 
at  41%  compound  interest  ? 

9.  What  principal  will  gain  $274.07131  in  5  years,  at 
5% ^compound  interest? 

ANNUITIES. 

374.  An  Annuity  is  a  specified  sum  of  money  payable 
at  equal  intervals  of  time. 


300  ARITHMETIC. 

Note.  We  shall  consider  in  tlie  present  chapter  those  cases  only 
in  which  the  payments  are  annual. 

The  Amount  or  Final  Value  of  an  annuity  is  the  sum  of 
all  the  payments,  together  with  the  interest  on  each  pay- 
ment from  the  time  it  becomes  due  until  the  annuity  ceases. 

The  Present  Worth  of  an  annuity  is  that  sum  of  money 
,  which,  at  the  specified  rate  of  interest,  will  amount  to  the 
final  value. 

375.  Annuities  at  Simple  Interest. 

Problems  in  annuities  at  simple  interest  may  be  solved 
by  aid  of  the  principles  of  arithmetical  progression. 

1.  Find  the  amount  of  an  annuity  of  $400  for  6  years, 
at  5%  simple  interest. 

The  first  payment  draws  interest  for  five  years ;  the  second  pay- 
ment for  four  years  ;  etc. 

Now  the  amount  of  the  first  payment  at  the  en^  of  five  ye-ars  is 
$400  X  1.25,  or  1 500. 

The  amount  of  the  second  payment  at  the  end  of  four  years  is 
$400  X  1.20,  or  $480;  etc. 

We  then  have  an  arithmetical  progression,  whose  first  term  is  $  500, 
last  term  $  400,  and  number  of  terms  6. 

Therefore,  by  Art.  366,  the  sum  of  the  terms  is 

.  f  X  ($500  +  $400),  or  $2700,  Ans. 
From  the  above  example,  we  derive  the  following 

RULE. 

Find  the  amount  of  the  annual  payment  for  a  number  of 
years  less  by  1  than  the  given  time. 

Add  this  amount  to  the  annual  payment,  and  multiply  the 
result  by  one-half  the  given  number  of  years. 

2.  Find  the  present  worth  of  the  annuity  of  Ex.  1. 

By  Art.  331,  the  present  worth  of  $2700  due  6  years  hence,  at  5  %, 
is  |2700^  or  $2076.92  +  ,  Ans. 


PROGRESSIONS.  301 

3.  What  annuity  to  continue  for  4  years,  at  6%  simple 
interest,  can  be  purchased  for  f  1090  ? 

An  annuity  of  $  1  to  continue  for  4  years,  at  6  %  simple  interest, 
will  amount  to  |  x  ($  1.18  +  $  1),  or  $4.36. 

The  amount  of  $  1090  for  4  years,  at  6  %,  is  ^  1351.60. 

Then,  an  annuity  of  as  many  dollars  can  be  purchased  for  $  1090 
as  4.36  is  contained  times  in  1351.60  ;  which  is  $310,  Ans. 

EXAMPLES. 

4.  Find  the  amount  and  present  worth  of  an  annuity  of 
$  300  for  5  years,  at  4%  simple  interest. 

5.  Find  the  amount  and  present  worth  of  an  annuity  of 
$  250  for  3  years,  at  4^%  simple  interest. 

6.  Find  the  amount  and  present  worth  of  an  annuity  of 
$800  for  8  years,  at  5%  simple  interest. 

7.  Find  the  amount  and  present  worth  of  an  annuity  of 
$720  for  7  years,  at  3f%.  simple  interest. 

8.  What  annuity  to  continue  for  6  years,  at  6%  simple 
interest,  can  be  purchased  for  $2070? 

9.  What  annuity  to  continue  for  8  years,  at  3%  simple 
interest,  can  be  purchased  for  $  4420  ? 

10.  What  annuity  to  continue  for  9  years,  at  4%  simple 
interest,  can  be  purchased  for  $  2088  ? 

11.  What  annuity  to  continue  for  11  years,  at  5^%  sim- 
ple interest,  can  be  purchased  for  $  5610  ? 

376.  Annuities  at  Compound  Interest.        ^ 

Problems  in  annuities  at  compound  interest  may  be  solved 
by  aid  of  the  principles  of  geometrical  pt-ogression. 

1.   Find  the  amount  of  an  annuity  of  $500  for  4  years, 
at  3%  compound  interest. 

The  fourth  or  last  payment  draws  no  Interest. 
The  amount  of  the  third  payment  at  the  end  of  one  year,  at  3  %,  is 
$  500  X  1.03. 


302  ARITHMETIC. 

The  amount  of  the  second  payment  at  the  end  of  2  years  at  3  % 
compound  interest  is  $500  x  (1.03)^  (Art.  373)  ;  etc. 

We  then  have  an  increasing  geometrical  progression,  whose  first 
term  is  $  500,  ratio  1.03,  and  number  of  terms  4. 

Therefore,  by  Art.  371,  the  sum  of  the  terms  is 

$500x[(1.03)^-11^  or  $2091.81  +  ,  Ans. 
.03 

2.  Find  the  present  worth,  of  th^  annuity  of  Ex.  1. 

By  Art.  373,  the  amount  of  $  1  for  4  years,  at  3  %  compound  inter- 
est, is  (1.03)4  dollars,  or  $1.12550881. 

Then  to  amount  to  $2091.81+  will  take  as  many  dollars  as 
1.12550881  is  contained  times  in  2091.81  +  ,  which  is  $1858.54  +  ,  Ans. 

3.  What  annuity  to  continue  for  3  years,  at  5%  compound 
interest,  can  be  purchased  for  ^  2522  ? 

An  annuity  of  $  1  to  continue  for  3  years,  at  5  %  compound  interest, 

will  amount  to  C^-^^^-  ^  dollars,  or  $3.1525. 
.05  '       ^ 

The  amount  of  $2522  for  3  years,  at  5%  compound  interest,  is 
$2522  X  (1.05)3,  or  $2919.53025. 

Then  an  annuity  of  as  many  dollars  can  be  purchased  for  $  2522,  as 
3.1625  is  contained  times  in  2919.53025  ;  which  is  $926.10,  Ans. 


EXAMPLES. 

4.  Find  the  amount  and  present  worth  of  an  annuity  of 
$200  for  3  years,  at  5%  compound  interest. 

5.  Find  the  amount  and  present  worth  of  an  annuity  of 
f  300  for  4  years,  at  6%  compound  interest. 

6.  Find  the  amount  and  present  worth  of  an  annuity  of 
$400  for  5  ye^s,  at  4%  compound  interest. 

7.  What  annuity,  to  continue  for  2  years,  at  6%  compound 
interest,  can  be  purchased  for  $  515  ? 

8.  What  annuity  to  continue  for  3  years,  at  4%  compound 
interest,  can  be  purchased  for  $  975.50  ? 

9.  What  annuity  to  continue  for  4  years,  at  5  %  compound 
interest,  can  be  purchased  for  $  600  ? 


MISCELLANEOUS  EXAMPLES.  303 

XXV.    MISCELLANEOUS  EXAMPLES. 
377.   1.   Find  the  value  of 


(7263  -  34242  -  439)  x  (61143  ^  837  -  748). 

2.  Divide  1661^  by  17,  and  reduce  the  result  to  a  mixed 
number. 

3.  Eeduce  f  If  to  91767ths. 

4.  A  man  spent  ^  of  his  money  for  provisions,  -f  of  the 
remainder  for  clothing,  -^-^  of  the  remainder  for  charity,  and 
had  f  9.10  left.     How  much  had  he  at  first  ? 

5.  Find  the  interest  of  $3528.75  from  Nov.  25,  1887,  to 
Sept.  11,  1891,  at  4i%. 

6.  Find  the  proceeds  of  a  3-months'  note  for  $  576,  dis- 
counted on  the  day  of  its  date  at  3f  %. 

7.  The  area  of  a  square  field  is  2  A.  77  sq.  rd.  17  sq.  yd. 
4|  sq.  ft. ;  find  its  side  in  rods,  yards,  and  feet. 

^    ^.      ,.,        396  X  425  X  1274 
»•   ^^^P^-^^y     4896x325x1078; 

9.  The  circumference  of  the  hind- wheel  of  a  carriage  is 
9  ft.  2  in.,  and  of  the  fore-wheel  7  ft.  9  in.  How  many 
times  does  each  wheel  turn  in  travelling  9  mi.  220  rd.  ? 

10.  Extract  the  square  root  of  .729275008576. 

11.  A  bin  8  ft.  3  in.  long,  5  ft.  8  in.  wide,  and  4  ft.  2  in. 
deep  is  filled  with  wheat.  If  a  bushel  is  equal  to  1^  cu.  ft., 
how  much  are  the  contents  worth,  at  96  cents  a  bushel  ? 

12.  Multiply  7  mi.  113  rd.  4  yd.  2  ft.  11  in.  by  27. 

13.  When  it  is  3.08  p.m.  at  St.  Petersburg,  Ion.  30°  19' 
48"  E.,  it  is  4  hr.  6&  min.  51|  sec.  a.m.  at  San  Francisco. 
What  is  the  longitude  of  San*  Francisco  ? 

14.  Simplify     64-5i  +  4t-3i 


304  ARITHMETIC. 

15.  A  can  do  a  piece  of  work  in  8i  hr. ;  A  and  B  together 
can  do  it  in  4^8_  ^j.^ .  ^nd  A  and  C  together  can  do  it  in  4  hr. 
How  many  hours  will  it  take  B  and  C  together  to  do  the 
work  ? 

16.  Subtract  y\%  from  i|f ,  and  reduce  the  result  to  its 
lowest  terms. 

17.  rind  the  exact  number  of  days  from  May  15,  1873, 
to  March  12,  1892. 

18.  A  gentleman  bequeathed  37|-%  of  his  property  of 
1 15120  to  his  wife,  44|%  of  what  remained  to  his  son,  71f  % 
of  the  balance  to  his  daughter,  and  the  remainder  to  charity. 
How  much  did  he  leave  to  charity  ? 

19.  How  long  must  $487  be  on  interest  at  3^%  to  gain 
$99.43? 

20.  Express  iff  as  a  circulating  decimal. 

21.  Find  the  side,  and  the  number  of  board  feet,  in  the 
squared  timber  that  can  be  sawed  from  a  log  whose  length 
is  19  ft.  7  in.,  and  diameter  at  the  smaller  end  16  in. 

22.  Multiply  83fff  by  59. 

23.  A  wheel  revolves  5000  times  in  travelling  8  mi.  296 
rd.     What  is  its  radius  in  inches  ? 

24.  Express  .008171875  as  a  common  fraction  in  its 
lowest  terms. 

25.  If  it  costs  $  98.55  to  plaster  a  hemispherical  dome 
whose  diameter  is  34  ft.  9  in.,  how  much  will  it  cost  to 
plaster  a  hemispherical  dome  whose  diameter  is  57  ft.  11  in.  ? 

26.  What  per  cent  above  cost  must  a  merchant  mark  an 
article,  in  order  to  be  able  to  sell  it  at  a  discount  of  16% 
from  the  list  price,  and  still  make  a  profit  of  11%  ? 

27.  Find  the  L.  C.  M.  of  1656,  3087,  and  8316. . 

28.  Divide  31  sq.  mi.  114  A.  132  sq.  rd.  21  sq.  yd.  3  sq. 

ft.  90sq.  in.  by  18. 


MISCELLANEOUS  EXAMPLES.  305 

29.  A  tradesman  sold  merchandise  for  $  1101.75,  and 
gained  |-f-  of  what  it  cost  him.  How  much  did  he  gain  by 
the  operation  ? 

30.  When  it  is  7.36  a.m.  at  Washington,  Ion.  77°  3'  37" 
W.,  what  time  is  it  at  Calcutta,  Ion.  88°  19'  2"  E.  ? 

31.  Divide  298i|  by  23. 

32.  A  train  of  54  cars  is  loaded  with  coal,  each  car  con- 
taining 4  long  tons,  18  long  hundred-weight.  What  is  the 
value  of  the  coal,  at  f  4.95  a  short  ton  ? 

33.  Find  the  amount  of  $  2893.40  from  March  27,  1885, 
to  March  18,  1889,  at  2i%. 

34.  I  paid  $  65,  including  $  1.75  for  the  policy,  for  insur- 
ing $  4600  on  a  house.     What  was  the  rate  of  insurance  ? 

35.  Express  8  mi.  289  rd.  3  yd.  2  ft.  11  in.  in  inches. 

36.  If  a  train  performs  a  certain  journey  in  5  h.  36  min., 
travelling  at  the  rate  of  56  feet  a  second,  how  long  will  it 
take  it,  travelling  at  the  rate  of  1085  yards  a  minute  ? 

37.  Find  the  present  worth  and  true  discount  of  $478.95 
due  1  yr.  9  mo.  10  d.  hence,  at  4%. 

38.  How  large  a  draft  on  Bremen  can  be  purchased  for 
$  500,  if  exchange  on  Germany  is  quoted  at  95}  ? 

39.  Divide  2J^  by  1|^,  and  reduce  the  result  to  a 
mixed  number. 

40.  Find  the  lateral  area  and  volume  of  a  pyramid  with 
a  square  base,  each  side  of  whose  base  is  26  in.,  and  whose 
altitude  is  84  in. 

41.  A  merchant  sold  goods  for  $  17212.50,  and  lost  if  of 
what  they  cost  him.     How  much  did  the  goods  cost  him  ? 

42.  A  rectangular  field  is  15  rd.  4  yd.  1^  ft.  long,  and 
12  rd.  3  yd.  1  ft.  wide.  How  much  is  it  worth  at  $592.90 
an  acre  ? 


306  AKITHMETIC. 

.  43.   Find  the  G.  C.  D.  of  54432,  63504,  and  98784. 

44.  Express  £  84  9s.  2d.  1  far.  as  a  fraction  of  £  123  Ss. 
9d.  3  far. 

45.  The  population  of  a  town- decreased  6^%  from  1870 
to  1880,  and  increased  13f  %  from  1880  to  1890.  If  the 
population  in  1890  was  4305,  what  was  the  population  in 
1870? 

46.  A  merchant  imported  875  sq.  yd.  of  rugs,  invoiced 
at  139.2  francs  a  square  yard.  What  was  the  duty,  at  55 
cents  a  square  yard,  and  45%  ad  valorem? 

47.  Prove  that  4057  is  a  prime  number. 

48.  Express  2-Hto"  ^^^  tA"  ^^  decimals,  and  divide  the 
first  result  by  the  second. 

49.  How  much  will  it  cost  to  cover  a  floor  20  ft.  3  in. 
long  and  15  ft.  7  in.  wide,  with  carpeting  28  in.  wide,  at 
$1.16  a  yard,  if  the  strips  run  lengthwise  of  the  floor? 
How  much  if  the  strips  run  across  the  room  ? 

50.  What  number  is  that  ^  of  |^  of  which  exceeds  ||- 
of  If  of  it  by  1^9^? 

51.  A  broker  receives  $  1000  to  invest,  after  deducting  a 
brokerage  of  1J%.  What  sum  can  he  invest,  and  what  is 
the  amount  of  his  commission  ? 

52.  Divide  1081  lb.  1  oz.  13  pwt.  9  gr.  by  2  lb.  0  oz.  16 
pwt.  3  gr. 

53.  At  what  rate  per  cent  will  $  540  amount  to  $  732.39 
in  6  y.  5  mo.  22  d.  ? 

54.  Eeduce  ^^Vtt  *^  ^^^  lowest  terms. 

55.  Eind  the  cost  of  a  pile  of  wood  29  ft.  7  in.  long,  5  ft. 
3  in.  high,  and  4  ft.  wide,  at  f  7.16|-  a  cord. 

56.  Simplify  ( 1.37 5  +  IJ  x  -^^  +  -^V  ^M^  ^^^  express 

\  .028        J 

the  result  as  a  decimal. 


MISCELLANEOUS  EXAMPLES.  307 

57.  If  a  bushel  of  wheat  weighs  59  lb.,  wh^^t  is  the  value 
of  nine  carloads  of  wheat,  each  weighing  8  T.  17  cwt.,  at 
$1.03|  a  bushel? 

58.  Simplify  (8^  -  4f3  )  _  (76  _  5f ) . 

59.  Extract  the  cube  root  of  456.266246971625. 

60.  A  bankrupt  owes  to  A,  $398.75;  to  B,  f  508.75;  to 
C,  $316.25;  and  to  D,  $563.75.  If  his  resources  are 
$  1115.40,  what  is  each  creditor's  share  ? 

61.  Find  the  G.  C.  D.  of  7429,  11339,  and  12673. 

62.  The  hypotenuse  of  a  right  triangle  is  28J  ft.,  and  one 
of  the  sides  about  the  right  angle  is  IJ  yd. ;  find  the  other 
side  in  inches. 

63.  If  a  tank  whose  length  is  6  ft.  5  in.  contains  578f| 
gallons  of  water,  what  is  the  length  of  a  similar  tank  which 
contains  2246^^-  gallons  ? 

64.  Express  158324  in.  in  terms  of  higher  denomi- 
nations. 

65.  An  agent  sells  415  yards  of  woollens,  at  $  1.52  a  yard, 
charging  2i%  commission.  He  invests  the  net  proceeds  in 
silks  at  $1.95  a  yard,  charging  3f%  commission.  How 
many  yards  can  he  buy  ? 

66.  A  can  do  a  piece  of  work  in  15  days,  B  in  18  days,  C 
in  21  days,  and  D  in  24  days.  How  many  days  will  it  take 
all  of  them  together  to  do  the  work  ?  If  they  receive  the 
sum  of  $79.95  for  the  work,  how  should  the  money  be 
divided  ? 

67.  Find  the  cost  of  a  draft  on  Chicago  for  $1500;  due  60 
days  after  sight,  with  interest  at  3|-%,  if  exchange  is  at  1J% 
discount. 

68.  If  five  men  can  do  a  piece  of  work  in  4  d.  5  h.  21 
min.  45  sec,  how  long  will  it  take  nine  men  to  do  the  work  ? 

69.  Find  the  exact  interest  of  $769.50  from  June  11, 
1891,  to  March  23,  1892,  at  ?>\%. 


808  ARITHMETIC. 

70.  Express  0.79839  lb.  troy  in  lower  denominations. 

71.  If  $14513.75  is  realized  from  the  sale  of  $17000 
worth  of  bonds,  including  a  brokerage  of  i%,  what  is  the 
quoted  price  of  the  bonds  ? 

72.  Find  the  last  term  and  the  sum  of  the  terms  of  the 
arithmetical  progression  ^,  -^,  if,  etc.,  to  63  terms. 

73.  Keduce  -^-f^j  to  its  lowest  terms. 

74.  Find  the  last  term  and  the  sum  of  the  terms  of  the 
geometrical  progression  1-J,  2|-,  4,  etc.,  to  10  terms. 

75.  Add  6  sq.  mi.  313  A.  152  sq.  rd.  21  sq.  yd.  8  sq.  ft. 
137  sq.  in.,  13  sq.  mi.  602  A.  67  sq.  rd.  14  sq.  yd.  3  sq.  ft. 
122  sq.  in.,  34  sq.  mi.  447  A.  112  sq.  rd.  9  sq.  yd.  5  sq.  ft. 
64  sq.  in.,  and  19  sq.  mi.  296  A.  89  sq.  rd.  28  sq.  yd.  7  sq.  ft. 
98  sq.  in. 

f     no)  What  sum  must  be  invested  in  a  6^%  stock  at  117, 
brokerage  i%,  to  yield  an  annual  income  of  $487.50  ? 

77.  Find  the  price  in  pounds,  shillings,  pence,  and  far- 
things, of  an  article  worth  $21.84,  if  the  sovereign  be  worth 

$4,871 

,  78.  A  sum  of  money  was  divided  between  A,  B,  C,  and 
D,  in  such  a  way  that  A  received  -^,^-^,C  ^^,  and  D  the 
remainder,  which  was  $  35.55.  What  was  the  sum  divided, 
and  how  much  did  each  receive  ? 

79.  If  a  bushel  =  1 J  cu.  ft.,  what  must  be  the  depth  of  a 
bin  5  ft.  4  in.  long,  and  4  ft.  9  in.  wide,  to  hold  98  bushels 
of  grain? 

^     80.    Express  f  ^  and  JJ)_9^  as  decimals,  and  find  the  prod- 
uct of  the  results. 

81.  Find  the  proceeds  of  a  note  for  S  875,  dated  Nov.  25, 
1892,  payable  60  days  after  date,  and  bearing  interest  at 
4%,  if  discounted  Dec.  11,  1892,  at  5i%. 


MISCELLANEOUS   EXAMPLES.  309 

82.  Find  the  cost  of  a  draft  on  Boston  for  $  2875,  due  90 
days  after  sight,  with, interest  at  5%,  if  exchange  is  at  2|% 
premium. 

83.  If  a  man  can  do  a  piece  of  work  in  8|-  days,  working 
10  h.  55  min.  a  day,  how  many  days  will  it  take  him,  work- 
ing 8  h.  36  min.  a  day  ? 

/  84.    At  what  rate  per  cent  will  ^1125  gain  $198.75  from 
Sept.  20,  1885,  to  June  5,  1890  ? 

'  85.  A  merchant  sold  goods  for  $658.35,  losing.  21f%  of 
what  they  cost  him.  At  what  price  should  the-  goods  have 
been  sold  so  as  to  gain  13^%  ? 

86.  What  is  the  duty,  at  40%  ad  valorem,  on  an  importa- 
tion of  crockery,  invoiced  at  £  896  5s.  6d.,  if  the  pound  ster- 
ling be  valued  at  $  4.8665  ? 

87.  Express  225  rd.  4  yd.  2  ft.  6f|  in.  as  a  fraction  of  a 
mile. 

88.  A  gentleman  left  .3  of  his  property  to  his  wife,  .4 
of  the  remainder  to  his  son,  .65  of  the  remainder  to  his 
daughter,  and  the  balance,  $  1543.50,  to  charitable  institu- 
tions.    How  much  did  each  receive  ? 

89.  How  many  shares  of  stock,  at  a  premium  of  13|^%, 
can  be  bought  for  $93854.25,  including  a  brokerage  of  i%  ? 

90.  What  is  the  equated  time  of  paying  $  519  due  in  30 
days,  $  348  due  in  60  days,  $  497  due  in  90  days,  and  $286 
due  in  120  days  ? 

91.  Arrange  in  order  of  magnitude  |-|,  -f-f,  and  i|-|-. 

92.  How  long  must  $  926.50  be  on  interest  at  21%  to 
amount  to  $1200? 

93.  Divide  35  d.  17  h.  41  min.  59  sec.  by  2.84. 

94.  How  many  cubic  feet  are  there  in  a  tapering  column 
17  ft.  in  height,  whose  lower  base  is  a  square  16  in.  on  a 
side,  and  upper  base  14  in.  on  a  side  ? 


310  ARITHMETIC. 

95.  Subtract  39if  f  from  57^^^^,  and  reduce  the  result  to 
its  lowest  terms. 

96.  Find  the  cost  of  384  boards,  each  12  ft.  10  in.  long, 
7  in.  wide,  and  |  in.  thick,  at  1 18.75  per  M. 

97.  The  local  time-  at  two  places  on  the  equator  differs 
by  9  h.  37  min.  33  sec.  What  is  the  distance  between 
them  in  miles,  if  a  degree  of  the  equator  be  taken  as  69^ 
miles  ? 

98.  Find  the  L.  C.  M.  of  4199,  7429,  and  12673. 

99.  If  a  cubic  foot  of  water  weighs  1000  oz.,  and  a  cubic 
inch  of  gold  weighs  ij  lb.,  what  is  the  specific  gravity  of 
gold  ? 

100.  Find  the  cost  of  938  shares  of  railway  stock  at  125f, 
brokerage  i%. 

101.  A  tank  contains  8  cu.  yd.  21  cu.  ft.  1048  cu.  in.  of 
water.  If  it  can  be  emptied  by  a  pipe  in  1  h.  24|  min.,  how 
many  cubic  inches  pass  through  the  pipe  in  one  second  Z 

102.  Divide  87843  into  parts  proportional  to  li,  2i  3J, 
^,  and  5i. 

103.  Find  the  amount  and  present  worth  of  an  annuity 
of  f  540  for  8  years,  at  4J%  simple  interest. 

104.  Extract  the  square  root  of  -f^  to  six  places  of  deci- 
mals. 

^105.  How  much  above  cost  must  a  tradesman '  mark  an 
article  costing  ^  24.64,  so  as  to  be  able  to  sell  it  at  a  dis- 
count of  8|^%  from  the  list  price,  and  still  make  a  profit  of 
15f  %  ? 

106.  1^  of  3iff  is  1^  times  what  number  ? 

107.  What  is  the  length  of  the  longest  straight  line  that 
can  be  drawn  on  a  square  floor  whose  area  is  237  sq.  ft 
97  sq.  in.? 


MISCELLANEOUS   EXAMPLES  311 

108.  Express  83|f|-  lb.  troy  in  grains. 

109.  What  will  be  the  face  of  a  draft,  due  30  days  after 
sight,  with  interest  at  41%,  which  can  be  bought  for  $1000, 
when  exchange  is  at  a  discount  of  3^%  ? 

iin     a-       ^^f     5.3-2.45  +  7.83 
^^  110.    Simplify 


1.613  --  6.05 


^  111.    Subtract  7  times  £  136  19s.  lOd  1  far.  from  12  times 
£  87  6s.  3c?.  21  far. 

112.  What  is  the  average  time  of  paying  $  186  due  March 
12,  $  155.25  due  April  7,  $  414  due  May  29,  and  $  258.75  due 
July  20  ? 

•f(il^   What  principal  at  5%  interest  will  amount  to  $  600 
iiriry.  11  mo.  5  d.? 

114.  If  18|%  is  gained  by  selling  goods  for  $126.54, 
what  per  cent  would  be  lost  by  selling  them  for  $  76.96  ? 

115.  What  must  be  the  face  of  a  note  due  90  days  hence, 
which,  when  discounted  at  51%,  will  yield  $850  ? 

116.  Express  3  lb.  9  oz.  17  pwt.  14  gr.  in  avoirdupois 
weight. 

^  117.    Simplify  1?-— . 

9+  — 
^13 

118.  What  is  the  compound  interest  of  $1260  for  9  mo. 
at  7%,  interest  being  compounded  quarterly  ? 

119.  What  is  the  face  of  a  sight  draft  which  can  be  bought 
for  $869.35,  when  exchange  is  at  a  premium  of  |%? 

120.  If  a  tank  6  ft.  9  in.  long,  3  ft.  4  in.  wide,  and  2  ft. 
2  in.  deep,  holds  365f  gallons  of  water,  how  deep  must  a 
tank  be  that  is  5  ft.  10  in.  long,  and  3  ft.  9  in.  wide,  to  hold 
437^-  gallons  of  water  ? 


312  AHITlIiMETlC. 

.095 -.0005        3.42 -.006 


121.    Simplify 


.0087  -  .3        .000812  -  .04 

122.  I  bought  5  A.  136  sq.  r.d.  of  land  at  the  rate  of  $  1200 
an  acre,  and  sold  it  at  a  profit  of  21%.  At  what  price  per 
square  foot  did  I  sell  it  ? 

123.  Find  the  G.  C.  D.  of  62^2.,  55ff,  47|-ff,  and  633%. 

124.  A  town  whose  taxable  property  is  valued  at 
$  975400,  wishes  to  raise  f  14130.25.  There  are  479  polls, 
each  paying  $1.50.  What  is  the  rate  of  taxation?  What 
tax  will  be  paid  by  an  individual  who  i)ays  for  3  polls,  and 
has  taxable  property  to  the  amount  of  $  8600  ? 

125.  If  a  bushel  contains  2150.42  cu.  in.,  what  must  be 
the  depth  of  a  cylindrical  measure  12  in.  in  diameter,  to 
hold  a  peck  ? 

126.  What  principal,  at  6%  compound  interest,  will  gain 
$325  in  2  y.  2  mo.,  interest,  being  compounded  semi- 
annually ? 

127.  Multiply  together  2^«,  2^\,  2J«|,  and  2J||. 

128.  Express  -^-^  sq.  mi.  in  lower  denominations. 

129.  Find  the  cost  of  a  draft  on  Paris  for  2016.80  francs, 
exchange  on  France  being  quoted  at  5.17^^. 

130.  If  the  weight  of  a  cubic  foot  of  water  is  62i  lb.,  how 
many  cubic  inches  of  mercury  (specific  gravity  13.596) 
does  it  take  to  weigh  103  lb.? 

131.  Express  .07593  cd.  in  cubic  inches. 

132.  Extract  the  cube  root  of  j^  to  four  places  of  deci- 
mals. 

133.  A  triangular  plot  of  ground  contains  2.875  A.  If 
its  altitude  is  25^  rods,  what  is  its  base  in  feet  ? 

134.  Express  11  oz.  13  dr.  as  a  decimal  of  a  pound  troy. 

135.  Express  5  lb.  13  oz.  7  dr.  in  troy  weight. 


MISCELLANEOUS  EXAMPLES.  313 

136.  What  principal  at  1\%  interest  will  gain  $52.75 
from  Feb.  8,  1888,  to  Jan.  4,  1892  ? 

y  137.    What  common  fraction  will  produce  .38076923  ? 

138.  Express  9  cu.  ft.  771.3792  cu.  in.  as  a  decimal  of  a 
cord. 

139.  How  many  spherical  bullets,  each  \  in.  in  diameter, 
can  be  formed  from  five  pieces  of  lead,  each  in  the  form  of 
a  cone,  whose  altitude  is  21  in.,  and  radius  of  base  4  in.  ? 

140.  Find  the  L.  C.  M.  of  ^-^^Q^,  J?//^,  -fi^^^  and  \\^. 

141.  A  provision  dealer  uses  a  false  measure  of  3  pk. 
7  qt.  instead  of  a  bushel.  What  per  cent  does  he  gain  by 
his  dishonesty  ?     What  per  cent  do  his  customers  lose  ? 

142.  How  much  will  be  realized  from  the  sale  of  354 
shares  of  mining  stock  at  a  discount  of  27-|-%,  brokerage 

143.  A  man  travelled  69f  miles  the  first  day,  and  on  each 
succeeding  day  three-fifths  as  many  miles  as  on  the  next 
preceding.  How  far  did  he  travel  on  the  8th  day  ?  How 
far  did  he  travel  in  8  days  ? 

144.  If  69.84  pounds  of  cofPee  can  be  bought  for  %  30.55^, 
how  many  pounds  can  be  bought  for  %  76.89|-  ? 

145.  A  merchant  owes  $2143.50  due  in  10  months.  If 
he  pays  $425  in  3  months,  $580  in  5  months,  and  $278.50 
in  8  months,  when  should  he  pay  the  balance  ? 

146.  If  ^  of  the  price  received  for  an  article  is  gain, 
what  is  the  gain  per  cent  ? 

147.  Multiply  11  T.  17  cwt.  81  lb.  9  oz.  13  dr.  by  ||. 

148.  If  a  gallon  contains  231  cu.  in.,  what  must  be  the 
diameter  of  a  cistern  whose  depth  is  9  ft.,  to  hold  1080 
gallons  ? 

149.  Find  the  cost  of  a  bill  of  exchange  on  Liverpool  for 
£  224  16s.,  when  exchange  on  England  is  quoted  at  4.86}. 


314  ARITHMETIC. 

150.  If  the  pint  liquid  measure  contains  28.875  cu.  in., 
how  many  cubic  feet  are  there  in  a  barrel  of  31|^  gallons  ? 

-  151.   Extract  the  sixth  root  of  177210755.074809. 

152.  A  circular  garden,  85  ft.  in  diameter,  is  surrounded 
by  a  walk  6  ft.  wide.  How  many  square  yards  are  there  in 
the  walk  ? 

153.  If  30  men  can  dig  a  trench  108  feet  long,  8|  feet 
wide,  and  9  feet  deep,  in  10|^  days  of  6|  hours  each,  how 
many  days  of  8  hours  each  will  it  take  24  men  to  dig  a 
trench  96  feet  long,  12|-  feet  wide,  and  12  feet  deep  ? 

154.  For  what  amount  must  a  vessel  worth  $  12325,  and 
her  cargo  worth  $  8709.36,  be  insured  at  4J%,  in  order  that, 
in  case  of  loss,  the  owner  may  recover  the  value  of  the 
vessel  and  cargo,  and  the  premium  ? 

155.  A,  B,  and  C  formed  a  partnership  for  one  year.  A 
put  in  $-925,  and  at  the  end  of  5  months  added  $  250 ;  B  put 
in  $  1075,  and  at  the  end  of  10  months  withdrew  $  325 ;  G 
put  in  ^  1250,  and  at  the  end  of  8  months  withdrew  $  475. 
They  gained  $  1337.  What  was  each  partner's  share  of  the 
gain  ? 

156.  What  annuity  to  continue  for  13  years,  at  3J% 
simple  interest,  can  be  purchased  for  f  6292  ? 

157.  What  amount  is  due  April  2,  1891,  on  a  note  for 
$458,  dated  April  24,  1886,  with  4^%  interest  payable 
annually,  on  which  no  payments  have  been  made  ? 

158.  Find  the  lower  base  in  rods  of  a  trapezoid  whose 
area  is  66885  sq.  in.,  altitude  16^^  ft.,  and  upper  base  7|  yd. 

159.  Express  2  sq  mi.  in  square  inches. 

'  160.    Simplify      7^y;^_  .  X  Jf^  -  H- 
(^1  g-  —  ^5 ;       3-      oy  g  —  y  g- 

161.  Add  together  iff,  ifi,  m,  and  ^Vo^^  and  reduce 
the  result  to  its  lowest  terms. 


MISCELLANEOUS  EXAMPLES.  315 

162.  At  what  per  cent  below  par  must  a  4J%  stock  be 
quoted,  to  yield  the  same  per  cent  on  the  investment  as  a 
5|%  stock  at  a  premium  of  23|^%,  brokerage  in  each 
case  |-%  ? 

163.  Find  the  amount  and  present  worth  of  an  annuity 
of  $500  for  4  years,  at  5%  compound  interest. 

/  164.    Simplify  '-r-. — '-^  +  '- '—t-.,  and  reduce  the  result 

.69 +  .2      .5 -.246 
to  a  mixed  number. 

165.  A  grocer  bought  36  bu.  0  pk.  3  qt.  of  nuts,  at  $  3.20 
a  bushel,  and  sold  them  at  12  cents  a  liquid  quart.  If  a 
quart  liquid  measure  contains  57f  cu.  in.,  and  a  quart  dry 
measure  67^  cu.  in.,  did  he  gain  or  lose,  and  how  much  ? 

166.  Express  6  sq.  rd.  19  sq.  yd.  3  sq.  ft.  57  sq.  in.  as  a 
decimal  of  9  sq.  rd.  7  sq  yd.  2  sq.  ft.  12  sq.  in. 


167.    Simplify  (if  of  18f )  +  8f  -  4^ 
^    ^  4i-(344^14A)+U 


168.  If  the  specific  gravity  of  lead  is  11.4,  and  a  cubic 
foot  of  water  weighs  1000  oz.,  find  the  weight  in  pounds  of 
a  sphere  of  lead  11  in.  in  diameter. 

169.  What  amount  is  due  Oct.  7,  1892,  on  a  note  for 
$2725,  dated  Nov.  18,  1891,  and  bearing  interest  at  4%,  on 
which  the  following  payments  have  been  made:  Jan.  11, 
1892,  $520;  April  17,  1892,  $790;  June  25,  1892,  $655; 
and  Aug.  12,  1892,  $480? 

170.  A  man  sells  4i%  stock  to  the  par  value  of  $65400 
at  92 J,  and  invests  tlie  proceeds  in  3J%  stock  at  81|,  brok- 
erage i%  on  each  transaction.  Is  his  income  increased  or 
diminished,  and  how  much  ? 

--  171.  A  tapering  hollow  iron  column,  one  inch  thick,  is  24 
ft.  long,  10.  in.  in  outside  diameter  at  the  larger  end,  and  8 
in.  in  diameter  at  the  smaller.  Find  its  weight,  if  a  cubic 
inch  of  the  metal  weighs  .27  lb. 


316  ARITHMETIC. 

.     172.   Simplify  |^  +  ^-li,-(2.,.2H). 

173.  A  cistern  oa-n  be  filled  by  three  pipes  in  9  h.  20  mi n., 
8  h.  45  min.,  and  12  h.  36  min.,  respectively.  How  many 
hours  and  minutes  will  it  take  to  fill  the  cistern  if  all  the 
pipes  are  opened  together  ? 

174.  A  railway  embankment,  226  rods  in  length,  is  8  ft. 
6  in.  wide  at  the  top,  21  ft.  6  in.  wide  at  the  bottom,  and  6 
ft.  4  in.  high.  How  many  cubic  yards  of  earthwork  does  it 
contain  ? 

175.  On  a  note  for  $  1000,  dated  Feb.  24,  1887,  and  bear- 
ing interest  at  6%,  the  following  indorsements  were  made: 
Jan.  5, 1888,  $  175 ;  May  16, 1889,  $  30 ;  Dec.  8, 1889,  $  250; 
Aug.  28,  1890,  $  200.     How  much  was  due  March  19,  1891  ? 

176.  If  a  rail  weighs  75  pounds  to  the  yard,  how  many 
tons  of  rails  will  be  required  to  lay  a  piece  of  double  track 
railway  11  mi.  272  rd.  in  length  ? 

177.  A  train  leaves  A  for  B,  44  miles  distant,  travelling 
at  the  rate  of  a  chain  in  1-J-  seconds.  Eighteen  minutes 
later  a  train  leaves  B  for  A,  travelling  at  the  rate  of  |  of  a 
chain  a  second.     How  many  miles  from  B  will  they  meet  ? 

^178.    Simplify  (M  ^ /A)  +  (5|f-f- l^V). 
(6Hx3A)-(4Hx  2f) 

179.  A  room  22  ft.  4  in.  long,  15  ft.  9  in.  wide,  and  9  ft. 
6  in.  high,  has  three  doors,  each  3  ft.  wide  and  6  ft.  9  in. 
high,  four  windows,  each  3  ft.  wide  and  5  ft.  5^  in.  high,  and 
is  surrounded  by  a  base-board  9  in.  wide.  How  much  will 
it  cost  to  plaster  it,  at  42  cents  a  square  yard  ?  How  much 
will  it  cost  to  paper  it  with  paper  21  in.  wide,  10  yards  to 
a  roll,  at  $1.12  a  roll? 

180.  What  annuity  to  continue  for  5  years,  at  3%  com- 
pound interest,  can  be  purchased  for  $  1000  ? 


MISCELLANEOUS   EXAMPLES. 


317 


181.    Find  the  equated  time  for.  paying  the  balance  of 
the  following  account : 


Dr. 


William  Lewis.  • 


Cr. 


1889 

1889 

Oct.    29 

To  Mdse.  30  d. 

$500 

Dec.  20 

By  Cash. 

$750 

Nov.  24 

1.1,       It 

250 

1890 

1890 

Jan.  19 

"  Draft,  90  d. 

600 

Jan.     6 

"       2  mo. 

600 

Feb.     9 

"       "      1  mo. 

450 

Feb.  27 

"       "       GO  d. 

1000 

Mar.  28 

"  Cash. 

200 

MISCELLANEOUS    EXAMPLES   INVOLVING   THE   METRIC 
SYSTEM. 

378.   1.   Express  .00527*^^  of  water  in  cubic  centimeters, 
and  find  its  weight  in  dekagrams. 

2.  Divide  .0072321273'="'^'"  by  7.489,  and  express  the  re- 
sult as  a  decimal  of  a  cubic  millimeter. 

3.  Find  the  altitude  in  dekameters  of  a  trapezoid  whose 
area  is  149878^'^'='",  and  bases  .05787""'  and  29.65*^"',  re- 
spectively. 

4.  A  wood-pile  is  32  ft.  long,  4  ft.  wide,  and  5  ft.  6  in. 
high.     How  much  is  it  worth,  at  $  2.50  a  ster  ? 

5.  If  a  train  travels  at  the  rate  of  36  miles  an  hour, 
what  is  its  rate  in  hektometers  a  minute  ? 

6.  Find  the  weight  in  dekagrams  of  a  ream  of  paper, 
each  sheet  24*='"  long  and  15'='"  wide,  if  a  sheet  of  the  same 
thickness  125""™  long  and  8""  wide  weighs  8.2*^«. 

7.  Express  2  mi.  223  rd.  4  yd.  1  ft.  9  in.  in  kilometers. 

8.  How  much  do  I  lose  by  buying  5  A.  135  sq.  rd.  of 
land  at  $  300  an  acre,  and  selling  it  at  $  675  a  hektar  ?  ^ 

9.  Express  2  gal.  3  qt.  1  pt.  3  gi.  in  deciliters. 
10.   How  many  chains  are  there  in  a  hektometer  ? 


818  ARITHMETIC. 

11.  How  mauy  decimeters  are  there  in  a  link  ? 

12.  A  rectangular  garden,  35.4'"  long  and  .289"'"  wide,  is 
surrounded  by  a  wQ,lk  19.6*^'"  in  width.  Find  the  area  of 
the  walk  in  ars. 

13.  Express  .874^^  in  liquid  measure,  and  also  in  dry 
measure. 

14.  If  it  costs  $  9.54  to  trc.vel  397^  miles  by  rail,  what 
is  the  rate  of  fare  in  cents  per  kiiometer  ? 

15. ,  How  many  hektoliters  of  grain  can  be  put  into  a 
receptacle  4.38'"  long,  27.9^"^  wide,  and  185""  deep  ?  How 
many  myriagrams  of  water  can  be  put  into  it  ? 

16.  How  many  bricks,  each  2.3*^"'  long,  86'"™  wide,  and 
5.04*=""  thick,  will  be  required  to  build  a  wall  .645"™  long, 
.46™  wide,  and  1.89^™  high  ? 

17.  Find  the  value,  at  12  cents  a  square  meter,  of  a  circu- 
lar piece  of  land,  whose  circumference  is  one  hektometer. 

18.  How  many  silver  half-dollars  can  be  coined  from 
ten  bars  of  silver,  each  55*='"  long,  36'"™  wide,  and  25"""  thick, 
if  each  coin  weighs  12.5^,  and  the  specific  gravity  of  silver 
is  10.5  ? 

19.  A  merchant  imports  3500™  of  silk,  invoiced  at  6.5 
francs  a  meter,  and  sells  it  at  $  1.55  a  yard.  If  the  franc 
is  valued  at  19.3  cents,  how  much  does  he  gain  ? 

20.  If  a  cubic  foot  of  granite  weighs  167  lb.,  what  is  the 
weight  of  a  cubic  meter  in  metric  tons  ? 

21.  How  much  will  it  cost  to  cover  a  floor  6.68™  long  and 
5.84™  wide,  with  carpeting  78*=™  wide,  at  -75  cents  a  meter,  if 
the  strips  run  lengthwise  of  the  room  ?  How  much  if  the 
strips  run  across  the  room  ? 

22.  Find  the  length  in  hektometers  of  the  longest  straight 
line  that  can  be  drawn  in  a  rectangular  field,  whose  area  is 
14.44908^'^^™,  and  width  .02776^'". 


MISCELLANEOUS   EXAMPLES.  319 

» 

23.  Find  tlie  volume  in  cubic  decimeters  of  a  frustum  of 
a  square  pyramid,  whose  altitude  is  825""",  lower  base  .28™  on 
a  side,  and  upper  base  22. 6*"™  on  a  side. 

24.  How  many  yards  of  fence  will  be  required  to  enclose 
a  circular  grass-plot  whose  area  is  6.157336*  ? 

25.  A  wood-pile  is  8.2™  long,  14*^  wide,  and  165'""  high, 
rind  its  value  at  f  6.70  a  cord.  ' 

26.  A  cubical  tank  holds  563975.2°^  of  water ;  what  is  its 
depth  in  dekameters  ? 

27.  How  many  dekaliters  of  petroleum  (specific  gravity 
.8778)  does  it  take  to  weigh  84663.81"^  ? 

28.  What  is  the  depth  in  decimeters  of  a  cylindrical 
tank,  83.5'^'"  m  diameter,  which  holds  9856809.27*^-  of  water  ? 

29.  Express  9.38^*^  in  ordinary  square  measure. 

30.  A  ditch  is  142*^'"  long,  18.3'^"^  wide,  and  876'"'"  deep. 
How  many  metric  tons  of  water  will  it  contain  ? 

31.  A  rod  of  steel  (specific  gravity  7.8)  is  5.5'"  long,  and 
18™"*  in  diameter.     Find  its  weight  in  hektograms. 

32.  Express  15  lb.  9  oz.  17  pwt.  21  gr.  in  dekagrams. 

33.  A  tank  is  2.28™  long,  1.75'"  wide,  and  1.625™  deep. 
How  many  kilograms  of  oil  (specific  gravity  .898)  will  it 
contain  ? 

34.  What  is  the  surface  in  square  decimeters  of  a  sphere 
whose  volume  is  2572446.8^"™™  ? 

35.  A  cubical  piece  of  lead,  whose  entire  surface  is 
;[3  5sqdm^  is  melted,  and  formed  into  a  cone  the  radius  of 
whose  base  is  15*=™.     Find  the  altitude  of  the  cone  in  meters. 

36.  What  is  the  diameter  in  dekameters  of  a  cylindrical 
cistern  32^™  deep,  which  holds  769.692^'  of  water? 

37.  The  volume  of  a  cone  is  33250.6944^"''™,  and  its  alti- 
tude is  7.2*^™.  Find  the  radius  of  its  base  in  meters,  and 
its  lateral  area  in  square  dekameters. 


320  ARITHMETIC. 

38.  What  is  the  depth  in  millimeters  of  a  tank  that  is 
188'='"  long  and  12.5'^'"  wide,  and  holds  2.5949546'^  of  alcohol 
(specific  gravity  .791)? 

39.  How  many  cubic  centimeters  of  metal  are  there  in 
a  hollow  iron  tube  of  uniform  diameter,  whose  length  is 
3.2™,  thickness  2"'°,  and  outside  diameter  1.2*=™  ? 

40.  A  bar  of  aluminum,  '9.2*^'"  long  and  2.5'"^  in  diameter, 
weighs  11.6062485°^.     What  is  its  specific  gravity  ? 

41.  Express  .378^^  in  avoirdupois,  and  in  troy  weight. 

42.  If  a  cubic  decimeter  of  silver  weighs  104.93°^,  what 
is  the  weight  of  a  cubic  foot  in  pounds  avoirdupois  ? 

43.  A  river,  whose  current  flows  at  the  rate  of  6.4^™  an 
hour,  is  .16°™  deep  and  84.6°™  wide.  How  many  metric 
tons  of  water  pass  a  given  point  in  43  min.  20  sec.  ? 

44.  If  a  rail  weighs  36^^  to  the  meter,  how  many  pounds 
does  it  weigh  to  the  yard  ? 

45.  A  tank  containing  20457.2°^  of  water  has  two  taps. 
One  fills  it  at  the  rate  of  342'="'=™  a  second,  and  the  other 
empties  the  contents  at  the  rate  of  a  dekaliter  in  .9765625 
minutes.  How  many  hours  and  minutes  will  it  take  to  fill 
the  tank  if  both  taps  are  opened  ? 

46.  The  cross-section  of  a  tunnel  2.5^™  in  length,  is  in 
the  form  of  a  rectangle  4.2™  wide  and  3.8™  high,  surmounted 
by  a  semicircle,  whose  diameter  is  equal  to  the  width  of 
the  rectangle.  How  much  did  it  cost  to  excavate  it,  at  $  6 
a  cubic  meter  ? 

47.  A  cannon  is  in  the  form  of  a  frustum  of  a  cone, 
joined  to  a  hemisphere  whose  diameter  is  equal  to  that  of 
the  larger  end  of  the  cannon.  The  diameter  of  the  larger 
end  of  the  cannon  is  4.2^™,  and  of  the  smaller  end  32*'™ ;  and 
its  entire  length  is  2.1™.  The  interior  bore  of  the  cannon  is 
24em  ^j^  diameter,  and  2™  deep.  How  many  cubic  decimeters 
of  metal  were  used  in  its  construction  ? 


APPENDIX. 


321 


APPENDIX. 


Measures  of  Length. 
A  furloDg 
A  league 
A  fathom 


MEASURES. 

=    40  rods. 
=      3  miles. 
=      6  feet. 


A  cable-length  =  120  fathoms. 

A  7iautical  mile,  geographical  mile,  or  k7iot,  is  equal  to  1.15 
miles. 
A  marine  league  is  equal  to  3  nautical  miles. 


A  line 

=  yL  i^c^- 

A  hand 

=   4  inches. 

A  palm 

=    3  inches. 

A  span 

=    9  inches. 

A  cubit 

=  18  inches. 

Cloth  Measure. 

2\  inches 

=  1  nail. 

4  nails 

=  1  quarter  of  a  yard. 

4  quarters 

=  1  yard. 

3  quarters 

=  1  Ell  Flemish. 

5  quai'ters 

=  1  Ell  English. 

Measures  of  Area. 

A  rood 

=    40  square  rods. 

A  square 

=  100  square  feet. 

Measures  of  Volume. 

A  cord  foot 

=  16    cubic  feet. 

A  perch 

=  24f  cubic  feet. 

322  ARITHMETIC.' 

A  perch  of  masonry  is  16^  ft.,  or  1  rod  long,  IJ  feet  wide, 
and  1  foot  high. 

Measures  of  Weight. 

A  quarter  =  25    pounds  av„ 
A  stone      =  14    pounds  av. 
A  pig         =  21-1-  stone. 
A  fother    =    8    pigs. 

Diamond  Weight. 
4  quarters  =  1  grain. 
4  grains      =  1  carat. 

A  grain,  diamond  weight,  is  equal  to  four-lifths  of  a  grain 
troy,  and  a  carat  is  equal  to  3|-  grains  troy. 

Measures  of  Time. 

A  solar  year  is  the  time  required  for  the  sun,  after  leav- 
ing either  equinox,  to  return  to  it  again ;  and  is  365  days, 
5  hours,  48  minutes,  and  49.7  seconds. 

A  sidereal  year  is  the  time  required  for  the  earth  to  make 
a  complete  revolution  about  the  sun;  and  is  365  days,  6 
hours,  9  minutes,  and  9.6  seconds. 

A  common,  civil,  or  legal  year  is  one  of  365  days ;  and  a 
leap,  or  bissextile  year  is  one  of  366  days. 

By  the  Julian  Calendar,  instituted  by  Julius  Caesar,  every 
fourth  year  is  made  a  leap  year ;  which  produces  an  error, 
as  compared  with  the  solar  year,  of  44  minutes  and  41.2 
seconds  in  four  years,  or  about  one  day  in  129  years. 

By  the  Gregorian  Calendar,  instituted  by  Pope  Gregory 
XIII.,  in  1582,  those  years  only  are  leap  years  whose  num- 
bers are  divisible  by  4,  and  not  by  100,  unless  they  are  also 
divisible  by  400. 

The  error  of  the  Gregorian  Calendar  amounts  to  about 
one  day  in  3866  years. 


APPENDIX.  323 

The  reckoning  of  time  by  the  Julian  Calendar  is  termed 
Old  Style,  and  by  the  Gregorian  Calendar  New  Style. 

The  latter  is  used  by  most  civilized  nations,  and  was 
adopted  by  England  in  1752 ;  in  that  year,  by  Act  of  Par- 
liament, the  day  following  Sept.  2d  was  counted  as  Sept.  14. 

The  present  difference  between  the  two  styles  is  12  days. 

To  find  the  Difference  in  Time  between  Two  Dates. 

The  following  method  is  largely  used  by  business  men  : 

The  year  is  regarded  as  consisting  of  12  months  of  30 
days  each. 

The  months  are  represented  by  their  numbers;  thus,  Jan- 
uary is  the  first  month,  February  the  second,  etc. 

The  difference  in  time  is  then  found  as  in  subtraction  of 
denominate  numbers. 

Example.  Find  the  difference  in  time  between  Oct.  15, 
1883,  and  June  7,  1892. 

1892  6  7  June  is  the  sixth  month  of  the 

188S        10  15  year,  and  October  the  tenth  month. 

—  15  d.  from  37  d.  leaves  22  d. 

18  y.     7  mo.  22  d.,    Ans.  10  mo.  from  17  mo.  leaves  7  mo. 

1883  from  1891  leaves  8  y. 
Then  the  required  result  is  8  y,  7  mo.  22  d. 

In  some  States,  the  actual  "  umber  of  days  in  the  preceding 
month  must  be  used,  when  the  number  of  days  in  the  sub- 
trahend time  is  greater  than  the  number  in  the  minuend 
time. 

Thus,  in  the  above  example,  the  month  preceding  June 
has  31  days. 

AVe  should  then  say,  15  d.  from  38  d.  leaves  23  d. ;  and 
the  time  would  be  8  y.  7  mo.  23  d. 

Comparison  of  Thermometers. 

In  the  Fahrenheit  Scale  (F.),  the  temperature  of  the  freez- 
ing point  of  water  is  marked  32°,  and  the  temperature  of 
the  boiling  point  212° ;  the  difference  being  180°. 


324  ARITHMETIC. 

In  the  Centigrade  Scale  (C),  the  freezing  point  is  marked 
0°,  and  the  boiling  x)oiiit  100°. 

In  the  Reaumur  Scale  (R.),  the  freezing  point  is  marked 
0°,  and  the  boiling  point  80°. 

Thus,  1°  F.  =  ;  0  0,  or  1°  C,  and  j\%,  or  4°  R. 
r  C.  =  -;-|0-,  or  f  F,  and  3%  or  f  R- 
r  R.  =  \%o_,  or  1°  F.,  and  i^%o,  or  |°  C. 

1.  Express  59°  F.  in  the  Centigrade  scale,  and  in  the 
Reaumur  scale. 

59°  F.  is  27°  above  the  freezing  point. 

But  27°  F.  is  equal  to  27  x  |,  or  15°  C,  or  to  27  x  ^,  or  12°  R. 

Hence,  59°  F.  is  equivalent  to  15°  C,  or  to  12°  R.,  Aiis. 

2.  Express  35°  C.  in  the  Fahrenheit  scale,  and  in  the 
Reaumur  scale. 

35°  C.  is  35°  above  the  freezing  point. 

But  35°  C.  is  equal  to  35  x  |,  or  63°  F.,  or  to  35  x  |,  or  28°  R. 

Hence,  35°  C.  is  equivalent  to  95°  F.,  or  to  28°  R.,  A71S. 

3.  Express  —  18°  R.  in  the  Fahrenheit  scale,  and  in  the 
Centigrade  scale. 

—  18°  R.  is  18°  below  the  freezing  point. 

But  18°  R.  is  equal  to  18  x  f,  or  40p  F.,  or  to  18  x  f ,  or  22^°  C. 

Hence,  -  18°  R.  is  equivalent  to  -  8|°  F.,  or  to  -  22^°  C,  A71S. 

EXAMPLES. 

Express  each  of  the  following  in  the  Centigrade,  and  in 
the  Reaumur  scale : 

4.    77°  F.        5.   140°  F.       6.   8°  F.  7.    -34°F. 

Express  each  of  the  following  in  the  Fahrenheit,  and  in 
the  Reaumur  scale : 

8.   55°  C.        9.    70°  C.       10.    -12°.         11.    -25°C. 
Express  each  of  the  following  in  the  Fahrenheit,  and  in 
the  Centigrade  scale : 

12.   52°  R.      13.   25°  R.       14.    -  10°  R.     15.    -  22°  R. 


APPENDIX.  325 

Miscellaneous  Terms. 

A  sheet  of  paper  folded  into  : 

2  leaves,  forms  d^  folio  ; 

4  leaves,  forms  a  quarto,  or  4to ; 

8  leaves,  forms  an  octavo,  or  8vo ; 
12  leaves,  forms  a  duodecimo,  or  12mo; 
18  leaves,  forms  an  18mo  j 
24  leaves,  forms  a  24mo. 

MONEY  AND  COINS. 
United  States  Money. 

For  the  denominations  of  United  States  money,  see  Art. 
142. 

The  coins  of  the  United  States  are  as  follows  : 

Gold;  the  quarter-eagle,  half-eagle,  eagle,  and  double-eagle. 

Silver;  the  dime,  quarter-dollar,  half-dollar,  and  dollar. 

Nickel ;  the  five-cent  piece. 

Bronze ;  the  ceyit. 

Note.  The  coinage  of  gold  dollars,  gold  three-dollar  pieces,  and 
nickel  three-cent  pieces,  was  suspended  by  act  of  Congress,  approved 
Sept.  26,  1890. 

The  monetary  system  of  Canada  is  the  same  as  that  of 
the  United* States. 

English  Money. 

For  the  denominations  of  English  money,  see  Art.  154. 
The  coins  of  Great  Britain  are : 

Gold ;  the  half-sovereign  and  sovereign. 
Silver;    the   three-penny  piece,   six-pence,   shilling,  florin, 
half-crown,  and  crown. 

Copper ;  the  halfpenny  and  penny. 

Note.     The  silver  four-penny  piece  is  no  longer  coined. 


326  ARITHMETIC. 

The  value  of  the  pound  sterling  in  United  States  money 
is  ^4.8665. 

French  Money. 

100  centimes  (c.)=  1  franc.   (/) 

The  coins  of  France  are  : 

Gold;  the  Jive-franc,  ten-franc,  twenty-franc,  forty-franc, 
and  hundred-franc  pieces. 

Silver;  th^  franc,  two-franc,  and  Jive-franc  pieces,  and  the 
tiuenty-Jive  centime  awd  fifty-centime  pieces. 

Bronze ;  the  one-centime,  two-centime,  five-centime,  and  ten- 
centime  pieces. 

Note.    The  gold  hundred-franc  piece  is  called  a  Napoleon. 

The  monetary  systems  of  Belgium  and  Switzerland  are 
the  same  as  that  of  France. 

The  value  of  the  franc  in  United  States  money  is  19.3 
cents. 

German  Money. 

100  pfennigs  (p/.)  =  lmark.     (m.) 
The  coins  of  Germany  are : 

Gold ;  the  five-mark,  ten-mark,  and  twenty-mark  pieces. 
Silver ;  the   one-mark,   two-mark,  and   three-mark  pieces  ; 
and  the  twenty-pfennig  Sindfifty-2)fennig  pieces. 
Nickel;  the  five-pfennig  and  ten-pfennig  pieces. 
Bronze ;  the  one-pfennig  and  tico-pfe7inig  pieces. 
Note.     The  silver  three-mark  piece  is  called  a  Thaler. 

The  value  of  the  mark  in  United  States  money  is  23.8 
cents. 

The  following  table  gives  the  values,  in  United  States 
money,  of  Foreign  Coins,  as  proclaimed  by  the  Secretary  of 
the  Treasury,  Oct.  1,  1891 : 


APPENDIX. 


827 


Country. 

Monetary  Unit. 

Value  in 
U.  S.  Money. 

Argentine  Republic. 

Peso. 

^0.965. 

Austria- Hungary. 

Florin. 

.357. 

Belgium. 

Franc. 

.193. 

Bolivia. 

Boliviano. 

.723. 

Brazil. 

Milreis. 

.546. 

British  Possessions,  N.  A., 

except  Newfoundland. 

Dollar. 

1.00. 

Central  American  States ; 

Costa  Rica,  Guatemala, 

Honduras,  Nicaragua, 

Salvador. 

Peso. 

.723. 

Chili. 

Peso. 

.912. 

China. 

f  Tael,  Shanghai. 

\     "    Haikwan  (customs). 

1.068. 

1.189. 

Colombia. 

Peso. 

.723. 

Cuba. 

Peso. 

.926. 

Denmark. 

Crown. 

.268. 

Ecuador. 

Sucre. 

.723. 

Egypt. 

Pound  (100  piasters). 

4.943. 

Finland. 

Mark. 

.193. 

France. 

Franc. 

.193. 

German  Empire. 

Mark. 

.238. 

Great  Britain. 

Pound  sterling. 

4.8665. 

Greece. 

Drachma. 

.193. 

Hayti. 

Gourde. 

.965. 

India. 

Rupee. 

.343. 

Italy. 

Lira. 

.193. 

Japan. 

r  Yen  (gold). 
1    "    (silver). 

.997. 
.779. 

Liberia. 

Dollar. 

1.00. 

Mexico. 

Dollar. 

.785. 

Netherlands. 

Florin. 

.402. 

Newfoundland. 

Dollar. 

1.014. 

Norway. 

Crown. 

.268. 

Peru. 

Sol. 

.723. 

Portugal. 

Milreis. 

1.08. 

Russia. 

Rouble. 

.578. 

Spain. 

Peseta. 

.193. 

Sweden. 

Crown. 

.268. 

Switzerland. 

Franc. 

.193. 

Tripoli. 

Mahbub  of  20  piasters. 

.652. 

Turkey. 

Piaster. 

.044. 

Venezuela. 

Bolivar. 

.145. 

Note.  The  francs  of  Belgium,  France,  and  Switzerland,  the  mark 
of  Finland,  the  drachma  of  Greece,  the  lira  of  Italy,  and  the  peseta  of 
Spain,  have  all  the  same  value. 


328 


ARITHMETIC. 


The  crowns  of  Denmark,  Norway,  and  Sweden  have  all  the  same 
value. 

The  boliviano  of  Bolivia,  the  sucre  of  Ecuador,  the  sol  of  Peru,  and 
thejpesos  of  Costa  Rica,  Guatemala,  Honduras,  Nicaragua,  Salvador, 
and  Colombia,  have  all  the  same  value. 

LEGAL  RATES  OF  INTEREST. 

The  Legal  Rate  of  interest  is  the  rate  which  is  established 
by  law. 

The  following  table  gives  the  legal  rate  of  interest  in 
each  state  and  territory  of  the  Union. 

When  no  rate  is  mentioned,  the  legal  rate  is  that  given 
in  the  left-hand  column;  if  specified  in  writing,  any  rate 
not  exceeding  that  in  the  right-hand  column  is  legal. 


State. 

Rate. 

State. 

Rate. 

State. 

Rate, 

Alabama. 

8 

8 

Kentucky. 

6 

6 

Nevada. 

7 

Any. 

Arkansas. 

6 

10 

Louisiana. 

5 

8 

Ohio. 

6 

8 

Arizona. 

10 

Any. 

Maine. 

6 

Any. 

Oregon. 

10 

12 

California. 

10 

Any. 

Maryland. 

6 

6 

Pennsylvania. 

6 

6 

Connecticut. 

6 

6 

Massachusetts. 

6 

Any. 

Rhode  Island. 

6 

Any. 

Colorado. 

8 

Any. 

Michigan. 

7 

10 

South  Carolina. 

7 

8 

Dakota. 

7 

12 

Minnesota. 

7 

10 

Tennessee. 

6 

6 

Delaware. 

6 

6 

Mississippi. 

6 

10 

Texas. 

6 

10 

Florida. 

8 

10 

Missouri. 

6 

8 

Utah. 

10 

Any. 

Georgia. 

7 

8 

Montana. 

7 

Any. 

Vermont. 

6 

6 

Idaho. 

10 

18 

N.  Hampshire. 

6 

6 

Virginia. 

6 

6 

Illinois. 

7 

7 

New  Jersey. 

6 

6 

West  Virginia. 

6 

8 

Indian  Ter. 

6 

Any. 

New  Mexico. 

6 

Any. 

Washington. 

8 

Any. 

Indiana. 

6 

8 

New  York. 

6 

6 

Wisconsin. 

6 

10 

Iowa. 

6 

8 

North  Carolina. 

6 

8 

Wyoming. 

12 

Any. 

Kansas. 

6 

8 

Nebraska. 

7 

10 

Dist.  ofCol. 

6 

10 

SPECIAL  STATE  RULES  FOR  PARTLAL  PAYMENTS. 
The  Connecticut  Rule. 

When  at  least  one  yearns  interest  has  accrued  at  the  time  of 
a  payment,  and  in  the  case  of  the  last  payment,  follow  the 
United  States  Rule  (Art.  324). 


APPENDIX.  329 

When  less  than  a  year's  interest  has  accrued  at  the  time  of  a 
payment,  except  the  last,  find  the  amount  of  the  principal  for 
an  entire  year,  and  the  amount  of  the  payment  for  the  re- 
mainder of  the  year  after  it  is  made,  and  subtract  the  amount 
of  the  payment  from  the  amount  of  the  principal  for  a  new 
principal;  but  if  the  payment  is  less  thaii  the  interest  which  is 
due  at  the  time  that  it  is  made,  no  interest  is  allowed  on  the 
payment. 

Example.  A  note  for  ^  2000,  dated  Oct.  8,  1888,  and 
bearing  interest  at  6%,  had  the  following  indorsements  : 
March  2,  1889,  $500;  Aug.  18,  1890,  $20.  What  was  due 
Dec.  23,  1890  ? 

Solution. 


Principal, 

$2000.00 

Int.  for  1  yr.. 

120.00 

Amount,  Oct.  8,  1889, 

$2120.00 

Amount  of  1st  payment  to  Oct.  8, 1889,  7  mo.  6  d., 

518.00 

New  Principal,  Oct.  8,  1889, 

$1602.00 

Int.  for  1  yr.. 

96.12 

Amount,  Oct.  8,  1890, 

$1698.12 

2d  payment. 

20.00 

New  principal,  Oct.  8,  1890, 

$1678.12 

Int.  to  Dec.  23,  1890,  2  mo.  15  d.. 

20.98 

Amount  due,  Dec.  23,  1890, 

$1699.10 

In  the  above  example,  the  first  payment  is  made  less  than  a  year 
after  the  date  of  the  note. 

We  find  the  amount  of  the  principal  for  an  entire  year  to  be  $  2120. 

The  remainder  of  the  year  after  the  first  payment  is  made  is  7  mo. 
6  d.  ;  and  the  amount  of  the  first  payment  for  this  time  is  $  518. 

Subtracting  $  518  from  $  2120,  the  new  principal  is  $  1602. 

The  amount  of  this  principal  for  one  year  is  $  1698.12. 

The  second  payment,  being  less  than  the  interest  which  is  due  at 
the  time  that  it  is  made,  draws  no  interest ;  then  subtracting  $  20  from 
$  1698.12,  the  new  principal  is  $  1678.12. 

The  amount  of  this  principal,  to  Dec.  23,  1890,  is  $  1699.10. 


330  ARITHMETIC. 

The  New  Hampshire  Rule  for  Partial  Payments  on  a  Note, 
or  other  Obligation,  drawing  Annual  Interest. 

If  in  any  year,  reckoning  from  the  time  when  the  annual 
interest  began  to  accrue^  payments  have  been  made,  compute 
interest  on  them  to  the  end  of  the  year. 

Find  also  the  accrued  annual  interest  on  the  principal,  and 
any  simple  interest  which  may  be  due  upon  .  unpaid  annual 
interest^  at  the  end  of  the  year. 

Then  the  amount  of  the  payment,  or  payments,  is  subtracted 
from  the  amount  due  on  the  note  at  the  end  of  the  year. 

But  if  the  payment,  or  payments,  are  less  than  the  sum  of 
the  simple  and  accrued  annual  interests  due  at  the  end  of  the 
year,  no  interest  is  allowed  on  the  payments. 

In  such  a  case,  simple  interest  is  computed  on  the  balance  of 
interest  due,  unless  the  payment,  or  payments,  are  less  thayi  the 
simple  interest  due  on  unpaid  annual  interest,  in  which  case  the 
balance  of  simple  interest  draws  no  interest. 

Example.  A  note  for  $  2500,  dated  July  10,  1887,  and 
bearing  interest  at  6%,  had  the  following  indorsements: 
April  4,  1889,  $  600 ;  May  26,  1890,  $  100.  What  was  due 
Nov.  25,  1891  ? 

Solution. 

Principal,  f  2500.00 

1st  Ann.  Int.,  to  July  10,  1888,  150.00 

2d  Ann.  Int.,  to  July  10,  1889,  150.00 

Int.  on  1st  Ann.  Int.  to  July  10,  1889,  9.00 

Amount  due,  July  10,  1889, 

1st  payment,  April  4,  1889, 

Int.  on  1st  payment  to  July  10,  1889, 

New  principal,  July  10,  1889, 

3d  Ann.  Int.,  to  July  10,  1890, 

2d  payment, 

Bal.  of  Int.  due  July  10,  1890,  ^31.96 


$  600.00 
9.60 

$2809.00 
609.60 

$  2199.40 
131.96 
100.00 

APPENDIX.      •  331 

Bal.  of  Int.  due  July  10,  1890,  $  31.96 

Int.  to  Nov.  25,  1891,  2.64 

New  principal,  July  10, 1890,  2199.40 

4th  Ann.  Int.,  to  July  10,  1891,  131.96 
Int.  on  principal  from  July  10,  1891,  to  Nov.  25, 

1891,  49.49 

Int.  on  4th  Ann.  Int.,  to  Nov.  25,  1891,  2.97 
Amount  due,  Nov.  25,  1891,                                         f  2418.42 

In  the  above  example,  the  first  payment  is  made  during  the  second 
year  after  the  date  of  the  note. 

The  accrued  annual  interest  on  the  principal,  at  the  end  of  the 
second  year,  is  $  300 ;  and  the  simple  interest  due  upon  the  unpaid 
annual  interest  of  the  first  year  is  $  9. 

Hence,  the  amount  due  on  the  note,  at  the  end  of  the  second  year, 
is  $2809. 

The  first  payment  is  made  3  mo.  6  d.  before  the  end  of  the  second 
year. 

The  amount  of  $600,  for  3  mo.  6  d.,  is  $609.60. 

Subtracting  this  from  $  2809,  the  new  principal  at  the  end  of  the 
second  year  is  $  2199.40. 

The  second  payment  is  made  during  the  third  year  after  the  date 
of  the  note. 

The  annual  interest  due  on  the  principal  at  the  end  of  the  third 
year  is  $131.96.  f 

The  second  payment  being  less  than  this,  draws  no  interest ;  then 
subtracting  .$100  from  $  131.96,  the  balance  of  interest  due  is  $31.96. 

Simple  interest  is  then  reckoned  on  this  balance  to  Nov.  25,  1891, 
amounting  to  $2.64. 

The  annual  interest  due  on  the  principal  at  the  end  of  the  fourth 
year  is  $131.96;  and  the  simple  interest  due  on  this,  Nov.  25,  1891, 
is  $2.97. 

The  interest  due  on  the  principal  from  July  10,  1891,  to  Nov.  25, 
1891,  is  $49.49. 

Adding  the  last  five  sums  to  the  new  principal,  July  10,  1890,  the 
amount  due  Nov.  25,  1891,  is  $2418.42. 

The  Vermont  Rule. 

If  in  any  year,  reckoning  from  the  time  when  the  annual 
interest  began  to  accrue,  payments  have  been  made,  compute 
interest  on  them  to  the  end  of  the  year. 


332  •  arithmj:tic. 

The  amount  of  the  payments  is  then  applied  : 
Firstf  to  cancel  any  simple  interest  which  may  he  due  upon 
unpaid  annual  interest. 

Second,  to  cancel  the  accrued  annual  interest. 
Third,  to  reduce  the  priricipal. 

The  Vermont  Rule  is  the  same  as  the  first  three  parar 
graphs  of  the  New  Hampshire  Eule. 

Note.  At  the  option  of  the  teacher,  the  examples  given  under  the 
United  States  Rule  (Art.  324)  may  be  performed  by  the  Connecticut 
Rule,  the  New  Hampshire  Rule,  or  the  Vermont  Rule. 

TO  COMPUTE  INTEREST   ON  ENGLISH  MONEY. 

To  compute  interest  on  English  money,  reduce  the  shil- 
lings, pence,  and  farthings,  if  any,  to  the  decimal  of  a  pound, 
and  then  proceed  as  in  United  States  money. 

The  decimal  of  a.  pound  in  the  result  should  be  reduced 
to  shillings,  pence,  and  farthings. 

1.  Find  the  interest  of  £  83  13s.  9d.  for  3  y.  6  mo.,  at  5%. 

We  have,  £83  13s.  9d.  =  £83.6875. 

The  interest  of  £83.6875  for  3  y.  6  mo.  at  5%  is  £  14.6453125. 
Reducing  £.6453125  to  lower  denominations,  the  result  is  12s.  lOd. 
3.5  far. 

Hence,  the  required  interest  is  £  14  12s.  lOd.  3.5  far. ,  Ans. 

EXAMPLES. 
Find  the  interest  and  amount : 

2.  Of  £56  5s.  for  3  y.  11  mo.,  at  3J%. 

3.  Of  £  31  14s.  6d  for  8  mo.  18  d.,  at  4%. 

4.  Of  £  27  8s.  3d  for  5  mo.  25  d.,  at  6%. 

5.  Of  £40  19s.  2d.  1  far.  for  1  y.  1  mo.  10  d.,  at  41%. 

6.  At  what  rate  per  cent  per  annum  will  £  36  17s.  4d 
gain  £  3  9s.  Id.  2  far.  in  1  y.  6  mo.  ? 

7.  In  what  time  will  £  190  gain  £  12  2s.  Sd.,  at  3%  ? 

8.  What  principal  will  gain  17s.  7d.  2  far.  in  1  y.  2  mo. 


APPENDIX.  333 

BUSINESS  FORMS. 

Receipt  in  Full. 

^  271y%%.  Cincinnati,  July  3,  1891. 

Received  from  Henry  Clark  two  hundred  and  seventy-one 
T%r  dollars,  in  full  of  all  demands  to  date. 

Edward  H.  Perry. 
Receipt  on  Account. 

$  100.  Buffalo,  Nov.  12,  1890. 

Received  from  James  E.  Hoyt  one  hundred  dollars  on 
account. 

William  G.  Faxon. 

Bank  Checks. 

A  Check  is  a  written  order  addressed  to  a  bank  by  a  per- 
son having  money  on  deposit,  requesting  the  payment,  on 
presentation,  of  a  certain  sum  to  the  person  named  therein, 
or  his  order. 

Form  of  a  Bank  Check. 
$  73^.  iVew?  York,  Feb.  21,  1892. 

The  National  Park  Bank. 

Pay  to  J.  H.  Crocker,  or  order,  seventy-three  ^^  dollars. 

No.  815.  W.  E.  Martin  &  Co. 

A  Certified  Check  is  one  on  the  face  of  which  the  Cashier 
or  Paying  Teller  of  the  bank  has  written  the  word  "  Certi- 
fied," and  under  it  his  signature ;  the  bank  in  this  way 
guarantees  the  payment  of  the  check. 

Form  of  a  CeHiJied  Check. 
j^^.  .  ,^         Boston,  Sept.  \7,IS91. 

The  iflfg^iKet  J^^onal  nJ^ank. 


Pay  to  George  f^^^i^es^^  ordmfone  hundred  and  fifty 
three  y%  dollar^,  ft/^  ^>^    \  a     /> 
No.  349.  P 1  •       yV/  William  Breck 


334  ARITHMETIC. 

Certificates  of  Deposit. 

A  Certificate  of  Deposit  is  a  statement  made  by  a  bank, 
certifying  that  the  person  named  therein  has  deposited  in 
the  bank  a  specified  sum  of  money. 

It  is  often  used  in  place  of  a  draft  in  making  a  remit- 
tance. 

Form  of  a  Certificate  of  Deposit. 
f  250i%.  Philadelphia,  May  4,  1892. 

The  National  Bank  of  Commerce. 

David  A.  King  has  deposited  in  this  hank  two  hundred  and 
fifty  -f^Q  dollars,  to  the  credit  of  himself  payable  on  the  return 
of  this  certificate,  properly  indorsed. 

No.  1047.  F.  F.  Hill,  Cashier. 

SAVINGS  BANK  ACCOUNTS. 

A  Savings  Bank  receives  small  sums  of  money  on  deposit, 
paying  interest  therefor. 

Money  depositedjon  or  before  certain  specified  dates  draws 
interest  from  those  dates. 

Interest  is  computed  either  monthly,  quarterly,  or  semi- 
annually, on  the  smallest  balance  that  has  been  on  deposit  dur- 
ing the  entire  term  of  interest;  but  no  interest  is  allowed  on 
any  sum  which  is  withdrawn,  for  the  time  between  the  date 
of  its  withdrawal  and  the  date  of  the  last  dividend^  nor  is 
interest  allowed  on  fractional  parts  of  a  dollar. 

If  interest  is  not  drawn  when  due,  it  is  added  to  the  prin- 
cipal, and  draws  interest  as  a  new  deposit;  thus,  savings 
banks  pay  compound  interest. 

A  depositor  in  a  savings  bank  receives  a  bank-book,  in 
which  all  deposits  and  amounts  withdrawn  are  entered. 

He  may  usually  withdraw  his  entire  deposit,  or  any  por- 
tion of  it,  at  any  time  when  he  sees  fit ;  but  some  banks 
require  a  week's  notice  before  paying  money  to  a  depositor. 


APPENDIX. 


335 


Example.  In  a  certain  savings  bank,  money  deposited  on 
or  before  the  first  days  of  January,  April,  July,  and  Octo- 
ber, draws  interest  from  those  dates  at  4%  ;  the  interest 
being  payable  on  the  above  dates. 

A  depositor,  whose  balance  Jan.  1,  1891,  was  $152.43, 
deposited  on  Feb.  3,  1891,  $75,  on  May  20,  1891,  $30,  and 
on  Aug.  12,  1891,  $  100. 

He  drew  on  April  27, 1891,  $  46,  and  on  Oct.  7, 1891,  $  151. 

What  was  his  balance  on  Jan.  1,  1892  ? 

Solution. 


Dates. 

Deposits. 

Drafts. 

Interest. 

Balances. 

1891 

Jan.     1 

$  152.43 

Feb.     3 

$  75.00 

227.43 

April  1 

$1.52 

228.95 

"    27 

$46.00 

182.95 

May  20 

30.00 

212.95 

July    1 

1.82 

214.77 

Aug.  12 

100.00 

314.77 

Oct.     1 

2.14 

316.91 

"       7 

151.00 

165.91 

1892 

Jan.    1. 

1.65 

167.56 

On  making  the  deposit  Feb.  3,  the  balance  becomes  $  227.43. 

On  April  1,  interest  is  paid  on  the  smallest  balance  that  has  been  on 
deposit  since  Jan.  1. 

No  interest  being  paid  on  fractional  parts  of  a  dollar,  the  interest 
due  April  1  is  1  %  of  $  152,  or  $  1.52  ;  and  the  balance  becomes  $  228.95. 

On  April  27,  .$46  is  withdrawn,  and  on  May  20,  $30  is  deposited, 
making  the  balance  $212.95. 

On  July  1,  interest  is  paid  on  the  smallest  balance  that  has  been  on 
deposit  since  April  1,  which  is  $  182.95. 

1  %  of  $  182  is  $  1.82  ;  which  makes  the  balance  July  1  $214.77. 


336 


ARITHMETIC. 


Aug.  12,  $  100  is  deposited,  and  the  balance  jDecomes  $314.77. 

On  Oct.  1,  interest  is  paid  on  a  balance  of  $  214,  amounting  to  $  2. 14  ; 
which  makes  the  balance  Oct.  1  $316.91. 

Oct.  7,  $151  is  withdrawn,  leaving  $165.91. 

On  Jan.  1,  1892,  interest  is  paid  on  $165,  amoimtingto  $1.65  ;  and 
the  balance  due  on  that  date  is  $167.56,  Ans. 

SCALES  OF  NOTATION. 

In  the  ordinary  method  of  expressing  whole  numbers,  a 
figure  in  any  place  represents  a  number  ten  times  as  great 
as  if  it  stood  in  the  next  place  to  the  right. 

This  method  of  representing  numbers  is  called  the  Com- 
mon, or  Decimal  Scale  of  Notation;  and  the  multiplier  10  is 
called  the  Radix. 

It  is  possible,  however,  to  represent  numbers  by  taking  as 
a  radix  any  whole  number  except  1. 

The  following  table  gives  the  name  and  radix  of  each  of 
the  first  eleven  scales : 


SCALB. 

Radix. 

SCALB. 

Radix. 

Scale. 

Radix. 

10 
11 
12 

Binary 
Ternary 
Quaternary 
Quinary 

2 
3 
4 
5 

Senary 
Septenary 
Octary 
Nonary 

6 

7 
8 
9 

Decimal 

Undenary 

Duodecimal 

To  express  a  number  in  any  uniform  scale,  as  many  dis- 
tinct symbols  are  required  as  there  are  units  in  the  radix  of 
the  given  scale. 

Thus,  in  the  decimal  scale,  10  symbols  are  required;  in 
the  binary  scale,  2  symbols,  0  and  1 ;  in  the  ternary  scale, 
3  symbols,  0,  1,  and  2 ;  the  cipher  being  a  symbol  in  every 
scale. 

In  the  duodecimal  scale,  12  symbols  are  required,  and  the 
numbers  10  and  11  are  represented  by  the  symbols  t  and  e, 
respectively. 


APPENDIX.  837 

In  the  decimal  scale,  a  digit  in  the  second  place  represents 
tens ;  in  the  third  place,  squares  of  tens ;  in  the  fourth 
place,  cubes  of  tens  ;  and  so  on. 

Thus,  3548  represents 

3x10^  +  5x102  +  4x10  +  8. 

In  like  manner,  in  any  scale,  a  digit  in  the  second  place 
represents  so  many  times  the  radix ;  in  the  third  place,  so 
many  times  the  square  of  the  radix ;  and  so  on. 

Thus,  in  the  nonary  scale,  7524  represents 

7x9^  +  5x92  +  2x9  +  4 

Example.  Write  in  the  senary  scale  the  numbers  from 
1  to  19  inclusive  in  the  common  scale. 

The  symbols  used  in  the  senary  scale  are  0,  1,  2,  3,  4,  5. 

The  numbers  from  1  to  5  inclusive  are  expressed  in  the  same  way 
in  each  scale. 

The  number  6,  being  1  six  and  no  owes,  is  expressed  10. 

The  number  7,  being  1  six  and  1  one,  is  expressed  11 ;  etc. 

Kesult :  1,  2,  3,  4,  5,  10,  11,  12,  13,  14,  15,  20,  21,  22,  23,  24,  26, 
30,  31. 

To  change  from  the  Decimal  to  any  other  Scale. 

Example.     Change  77609  to  the  duodecimal  scale. 

12)  77609  The  radix  of  the  duodecimal  scale  is  12. 

12)6467     5  Dividing  77609  by  12,  the  quotient  is  6467, 

1  o\KQQ   1  i  ^^  ^       and  the  remainder  5. 
S  In  That  is,  77609  =  6467  x  12  +  5. 

1^)44  10  or  t  Dividing  6467  by  12,  the  quotient  is  538,  and 

^     ^  the  remainder  11. 

38^e5   AyiS.  That  is,  77609  =  (538  x  12  +  11)  x  12  +  5 

'  *  =  538  X  122  +  11  X  12  +  5. 

Dividing  538  by  12,  the  quotient  is  44,  and  the  remainder  10. 
That  is,  77609  =  (44  x  12  +  10)  x  122  +  11  x  12  +  5 
=  44  X  123  +  10  X  122  +  11  X  12  +  5. 
Dividing  44  by  12,  the  quotient  is  3,  and  the  remainder  8. 
That  is,  77609  =  (3  x  12  +  8)  X  123  +  10  x  122  +  11  x  12  +  5 
=  3  X  12*  +  8  X  128  +  10  X  122  +  11  X  12  +  5. 
Thus,  77609  is  expressed  by  38«e5  in  the  duodecimal  scale. 


ANSWERS. 


Note.    In  the  following  collection  of  answers,  all  those  are  omitted 
which,  if  given,  would  destroy  the  utility  of  the  example. 


Art.  26.  Page  11. 

18. 

4440. 

22. 

12. 

1.  12. 

19. 

3304. 

23. 

1855. 

2.  52. 

20. 

21424. 

24. 

54. 

3.  32. 

4.  94. 

5.  101. 

6.  179. 

7.  100. 

8.  153. 

9.  506. 

21.  291. 

22.  12154. 

23.  3907. 

24.  255245. 

25.  291146. 

Art.  50.  Page  23. 
2.  537. 

2. 
3. 
4. 
5. 
6. 

Art.  66. 
Pages  34,  35. 

15. 
72. 
34. 
18. 
64.  . 

Art.  39.  Page  17. 

3. 

10503. 

7. 

225. 

1.  834588. 

4. 

507. 

8. 

35. 

2.  10270000. 

5. 

983. 

9. 

12. 

3.  3900383. 

6. 

6039;  107, 

Rem. 

10. 

48. 

4.  4054799. 

8. 

5062. 

11. 

9. 

5.  2172500. 

9. 

42860. 

12. 

13. 

6.  4116328. 

10. 

84 ;  347206 

,  Rem. 

13. 

6. 

7.  228958488. 

11. 

319;  72,  Rem. 

14. 

4. 

8.  148732062. 

12. 

800013. 

15. 

16. 

9.  433403622. 

13. 

t)284. 

16. 

65. 

10.  37709424. 

14. 

86395. 

17. 

30. 

11.  15716910. 

15. 

6. 

18. 

216. 

12.  602025849. 

16. 

48. 

19. 

75. 

13.  229480867. 

17. 

13. 

20. 

385. 

14.  248644665. 

18. 

26. 

21. 

84. 

15.  512. 

19. 

3. 

22. 

21  feet. 

16.  598. 

20. 

8. 

23. 

96  ;  in  first,  5  ;  in 

17.  1326. 

21. 

242. 

second,  7. 

ACADEMIC   ARITHMETIC. 


24.  28  inches. 

16. 

1155. 

10. 

98||. 

25.  24. 

17. 

2808. 

11. 

69H. 

26.  56  square  rods. 

18. 

690. 

12. 

65||. 

27.  22  feet 

. 

19. 

660. 

13. 

91f|. 

20. 

2592. 

14. 

85. 

Art.  67. 

Page  37. 

21. 

8505. 

15. 

145tf. 

3.  43. 

22. 

17640. 

16. 

82/^. 

4.  23. 

23. 

42120. 

17. 

mu- 

5.  31. 

24. 

42840. 

18. 

53ie. 

6.  19. 

25. 

94500. 

7.  41. 

26. 

90. 

Art.  84,    Page  45 

8.  43. 

27. 

240  niinutes. 

6. 

¥/. 

9.  118. 

28. 

$1080. 

7. 

w- 

10.  161. 

29. 

252;  A 

,  28  times ; 

8. 

-w.     .    .   . 

11.  37. 

B,  21 

times  ;  C, 

9. 

'II--     " ' 

12.  295. 

18  times. 

13.  73. 

Art.  85.    Page  46 

14.  83. 

Art.  74. 

Page  41. 

6. 

-¥/• 

15.  79. 

2. 

5491. 

7. 

W- 

16.  89. 

3. 

11951. 

8. 

-W- 

4. 

17081. 

9. 

'-IF- 

Art.  68. 

Page  37. 

5. 

20677. 

10. 

^iP- 

2.   13. 

6. 

26071. 

11. 

-11^- 

3.  17. 

7. 

6877. 

12. 

"M-- 

4.  19. 

8. 

8303. 

13. 

^H-- 

5.  23. 

9. 

11.339. 

14. 

10. 

12493. 

15. 

Art.  73. 

Page  40. 

11. 

15457. 

16. 

^W^- 

2.  180. 

12. 

1 10837. 

17. 

82      • 

3.  252. 

13. 

485377. 

9  3 

4.  3300. 

Art.  75. 

Page  42. 

Art.  89.  ■  Pago  47 

5.  432. 

2. 

549072. 

7. 

1 

6.  330. 

3. 

3174045. 

8. 

i- 

7.  2310. 

9. 

u- 

8.   1026. 

Art.  83. 

Page  45. 

10. 

tV- 

9.  900. 

3 

3/3- 

11. 

If. 

10.  528. 

4. 

9. 

12. 

t\. 

11.  3872. 

5. 

14f^ 

13. 

ff 

12.  5568. 

6. 

4711. 

14. 

t\- 

13.   1296. 

7. 

35i|. 

15. 

V-. 

14.  688. 

8. 

mh 

16. 

f. 

15.  1008. 

9. 

46. 

17. 

i- 

ANSWERS. 

Art.  90.    Page  48.       17.  f ,  f ,  ^V-  31-  97|. 

2.  tV.  18-  M,  If,  f  32.  172/,. 


5.   10. 


Art.  97. 
Art.  95.    Page  51.  Pages  54,  55. 


6.  V.  2.  4f.  5.  tV 

7.  1.  3.  ^3^^.  6.  H- 

8.  3^^.  4.  Mf.  7.  f|. 

9.  V-.  ^-  tM-  10.  if. 

10.  ^.  6.  f§f.  13.  If. 

11.  H-  7-  m-  14.  M- 

15.  if. 

Art.  91.  Page  49.                   Art.  96.  16.  ^V 

2.  / ^.  Pages  52,  53.  17.  ^5^. 

3.  If.  3.  jif.  18.  6i|. 


^^f- 


4.  if.  ■  4.  if) 

5.  f  |.  5.  3f.  20.  l^V 

6.  If.  6.  m  21.  7if 

7.  AV  7.  IHH.  22.  4tVV 

8.  fi.  8.  |.  23.  9fi. 

9.  f|.  9.  f|.  24.  6fi. 

10.  -If.  10.  ^%.  25.  5if. 

11.  t\V  11.  Iff.  26.  llff. 

12.  5/^.  27.  8ff. 

Art.  94.  13.  7|.  28.   7ff. 

Pages  50,  51.  14,  ^jj^,  29    13_2^ 

15.  y-.  30.  9/^. 

16.  If.  31.  12|f. 

5.  lit,  iff,  HI .  17.  W-  33.  i. 

6.  -W,  If,  M-  18.  If.  34.  4^. 

7.  fi,  H,  M-  19.  20^^.  35.  2|f. 

8-  M»  H,  If-  20.  27i.  36.  3f. 

9-  H.  If.  M-  21.  471.  37.  93V 
10.  t¥o.  t¥o.t¥o't¥o.  22.  lW:f.  38.  6jV 
11-  fi,  W,  M,  U'  23.  21ff.  39.  I7/3V 

12.  tV'o.tVo.tVo.t^^o.  24.    10,V  -  40..  IS^Viy. 

13.  fit,  Hh  IM,  Iff.  25.  1911.  41.  26ff. 

Iff.  ,  26.  24ff.  42.  32^\. 

14.  Vo\S     M,     in.  27.  20^%%. 

Uh  ^%%-  28.  f  If .  Art.  99.    Page  56. 

16.  y\  is  greater  than  29.  f  |f .  7.   V-. 


A. 


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ACADEMIC  ARITHMETIC. 


10.  V- 

16.  2tV 

7.  Y. 

11.  V- 

16.  3f 

8.  if. 

12.  V. 

9.  if 

13.   12|. 

Art.  104.    Page  01.      ^^'  fl' 

14.  49|. 

15.  64J,. 

6.  If. 

8.  /.. 

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10.    ,^7- 

11.  ¥• 

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11.  ^^. 

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16.  480. 

13.  H- 

17.  275|. 

14.  H. 

18.  617i 

15.  2%- 

19.  561f. 

20.  1080. 

16.  i^. 

17.  ¥-• 

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Art.  101.    Page  58.      ^'^'  tI(J- 

18.  |. 

19.  }t. 

20.  If. 

21.  if. 

22.  f. 

6.  f. 

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19.  W- 

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6.   V. 

23.   V/. 

7.  |. 

8.  if- 

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25.  W. 

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Art.  107.    Page  64. 

XV.       g. 

11.  V. 

12.  1. 

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25.  If. 

4.  if 
6.  f. 

13.  \^ 

14.  H- 

15.  f. 

16.  M. 

17.  %«-. 

26.  if. 

27.  If. 

28.  ff. 

29.  ff. 

30.  2^*. 

31.  H. 

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36.  A. 

6.  ^^. 

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•         8.  ^K 

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18.  ^h- 

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19.  A. 

20.  W- 

12.  If. 

13.  ,%. 

21.   V-- 

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22.   18. 

15.  V. 

23.   *^. 

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Art.  103.    Page  59.      ^®-   at- 

•     11 
18.  j%. 

19.  y. 

9.  1|. 

39.  If. 

10.  A. 

20.  |. 

11.   f|. 

Art.  105.    Page  63.      21.  f^. 

12.  2f. 

4.  f. 

22.  If. 

13.  3f. 

5.  If. 

23.  A. 

14.  3H. 

6.  /t- 

24.  If 

ANSWERS. 

5 

26.  J^. 

8.  If,  ih  e. 

8.  $234. 

26.  J. 

9.  Uh 

9.  2|. 

10.  2\l 

10.   $750. 

Art. 

108. 

11.  f. 

11.  Hind-wheel,  560 ; 

Pagres  64,  65. 

12.  U. 

fore-wheel,  600. 

3.   V-. 

13.  f. 

12.  $25000. 

4.  -%*. 

14.   1144. 

13.  Carriage,  $.390  ; 

5-  If. 

16.  ft. 

horse,  $234. 

6.  -V-. 

16.  26^^. 

14.  2^^^  hours. 

7.  |. 

17.  If 

15.  3j. 

8-  tV 

18.  xVrV 

16.  5i|. 

9.  f. 

19.  V. 

17.  $450. 

10.  U- 

20.  ff^. 

18.  36. 

21.  Y-. 

19.  $2535. 

Art.  109. 

Page  65. 

22.  W. 

20.  216. 

2-  tV 

23.  M. 

21.  $594. 

3.  ii. 

24.  Hf 

22.  83/j. 

4.  M- 

25.  9H. 

23.  106f  feet. 

5.  A- 

26.  U. 

24.  32  cents. 

6.  -V- 

27.  ff. 

25.  $1200. 

7.  ^f^. 

28.  S9,%. 

26.  $87. 

8.  AV 

29.  n,  If,  If. 

27.  Second  class,  42  ; 

9-  ^ih- 

30.  If 

third,  48 ;  fourth, 

31.  ^\\- 

63. 

Art.  110. 

Page  66. 

32.  V^. 

28.  If  acre. 

2.  -3^-. 

33.  fi. 

29.  5J. 

3.  48. 

34.  3^3^. 

30.  37^  miles. 

4.  -V- 

35.  If. 

31.  $123. 

5.  -y/-. 

36.  U. 

32.  $5400. 

6.  -W. 

37.  -Vt^-. 

33.  9|. 

7.  ^fi 

38.  H. 

34.  2f. 

8.  V/-. 

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35.  10. 

9.  W. 

40.  ^%. 

36.  26. 

41.  Iff. 

37.  U- 

Art. 

111. 

42.  If. 

38.  3|. 

Pages  66-68. 

39.  44  days. 

1.  leofff 

Art.  112. 

40.  11. 

2.  wf . 

Pages  69-73. 

41.  37f. 

4.  104|. 

4.  4if. 

42.  If 

5.  ^. 

5.  5jV 

43.  3f. 

6.  /r. 

6.  13^. 

44.  32. 

7.  ^^i^. 

7.  .$204. 

45.  402rV  feet. 

ACADEMIC  ARITHMETIC. 


46. 

163^  minutes; 

1  A 

Art.  119. 

Art.  127.    Page  83. 

will   have    gone 

Pages  76,  77. 

7. 

.102. 

around  20  times, 

2. 

27.3709G. 

8. 

7480. 

B  21  times";  i 

and 

3. 

38.65775. 

9. 

.936. 

C  25  times. 

4. 

225.17578. 

10. 

1.63. 

47. 

82^^  miles. 

5. 

11.20697. 

11. 

.897. 

48. 

3rV  days. 

6. 

5.742231. 

12. 

.00359. 

49. 

-H  rod. 

7. 

82.451502. 

13. 

62510. 

50. 

M- 

8. 

901.186486. 

14. 

.3731. 

51. 

3A. 

9. 

5341.02012. 

15. 

.00587. 

52. 

4|. 

16. 

2.35. 

53. 

44f  feet. 

Art.  120.    Page  77. 

17. 

45.6. 

54. 

The  sum  divided 

3. 

4.8536. 

18. 

50.83. 

was  1252  ;  A 

re- 

4. 

.25673. 

19. 

73.32. 

ceived    .$105, 

B 

6. 

.01054. 

20. 

.9225. 

$33|,  and  C$48. 

7. 

9.26771. 

21. 

490300. 

55. 

10|. 

8. 

.0037881. 

22. 

6812.5. 

56. 

3f  1  days  ;  A  per- 

10. 

2.44671188. 

23. 

.003961. 

forms  f^,  B 

Mr 

11. 

781.7513. 

C  i|,  and  D 

bo' 

12. 

.0765716. 

Art.  131.    Page  84. 

Art.  121.    Page  79. 

5. 

.325. 

3. 

242.311. 

7. 

.02531. 

Art.  118.    Page  76. 

4. 

.0541926. 

11. 

.000132. 

5. 

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1. 

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6. 

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2. 

jhs- 

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Art.  132. 

3. 

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8. 

62.8245. 

Pages  85,  86. 

4. 

hU- 

9. 

.0945162. 

4. 

.08. 

5. 

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.20727042. 

5. 

.6875. 

6. 

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11. 

.01088352. 

8. 

.01176. 

7. 

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9. 

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8. 

tVV. 

13. 

13.539. 

10. 

.078125. 

10. 

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rfk. 

15. 

4.9298795. 

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.96875. 

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.00076075064. 

13. 

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13. 

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Art.  125.    Page  80. 

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j\'u' 

6. 

1219.86. 

18. 

.85185. 

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19. 

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18. 

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4320.06. 

20. 

.07209. 

19. 

lUh- 

15. 

1431894.5. 

21. 

1.58416. 

20. 

w- 

16. 

143602200. 

22. 

.55008. 

answp:rs. 

7 

Art.  136.    Page  87. 

9. 

6.1. 

Art.  146. 

4.  .513. 

10. 

14600. 

Pages  92,  93. 

5.  .124. 

11. 

14. 

7.  .$226,463. 

6.   .684. 

12. 

.202. 

8.  $13.7582. 

7.   19.7227. 

13. 

.00228. 

9.  ^5.785,  ' 

8.  .32074. 

14. 

15.42. 

10.  275.42  c. 

9.  .46125. 

15. 

1.652. 

11.  $141,687. 

10.  .84653. 

16. 

.0784. 

12.   .0397377  e. 

11.  .00692307. 

17. 

782.88. 

17.  .127850.2. 

12.  .6421296. 

18. 

12.8464. 

18.  $3869.532. 

13.  7.1984126. 

19.  ■'$89..537022. 

20.  2928.076  d. 

Art.  140. 

21.  •'5*3.58. 

Art.  137.    Page  88. 

Pages  89,  90. 

22.  $0.39. 

3.  M. 

1. 

18.305665. 

23.  30.8. 

4-  U- 

2. 

.004486873. 

24.   .732. 

5.  tV 

6.  if 

3. 

.v^. 

25.  4.37. 

5. 

.615625. 

26.  .0688. 

7.  2|f. 

6. 

.0136. 

8-  i|. 

7. 

.28621. 

Art.  147. 

9.  tWt- 

8. 

tVo- 

Pages  94-97. 

10.  48ff. 

9. 

.061464585. 

3.  $21.60. 

11.  n. 

10. 

.03378. 

4.  $24.50. 

12.  t¥(jV 

11. 

3.3. 

5.  $11.76. 

13.  tV^. 

12. 

.0390625. 

6.  291. 

u.  laf. 

13. 

6245.8. 

t  $1.68. 

15.  m- 

14. 

28500. 

8.  $105. 

16.  7|tf|. 

15. 

ifT- 

9.  $728.18. 

17.  fH. 

17. 

.14087. 

10.  $12.07^. 

18.  Ul 

18. 

.000868. 

11.  $4149.03. 

19.  tffi 

19. 

.025587. 

12.  $9.21. 

20.  Mi 

20. 

.123456790. 

13.  .$60.48. 

21.  ^dlh- 

21. 

.09671875. 

14.  36,  and  21  gallons 

22.  ^Vj. 

22. 

90.38. 

remaining. 

23. 

m- 

15.  75. 

Art.  139. 

24. 

^'oV 

16.  $37.50. 

Pages  88,  89. 

25. 

.0459. 

17.  21  f. 

4.  21. 

26. 

-i. 

18.  .$75.55. 

6.   .262. 

27. 

If. 

19.  $120.75. 

6.  .175. 

28. 

H- 

20.  $6.66|. 

7.  410. 

29. 

.006. 

21.  .0083. 

8.  .05875. 

30. 

tVV. 

22.  .$5.25. 

ACADEMIC   ARITHMETIC. 


23. 

A,  $7.25 ; 

48. 

A,  $12.75; 

B,  $4.64  ; 

B,  $24.75  ; 

C,  $5.51. 

C,  $16.50. 

24. 

$1.44. 

49. 

lA- 

25. 

4.6903575  miles. 

26. 

$88.75. 

Art.  161. 

27. 

1  of  $17.67  is  the 

Pages  104,  105. 

greater  by  $.62. 

5. 

6720  pt. 

28. 

$17.46. 

6. 

6480  pwt. 

29. 

$35.18. 

7. 

6930  in. 

30. 

231. 

8. 

1105920  cu.  in. 

31. 

$974.05. 

9. 

4088  gi. 

32. 

141. 

10. 

7476  far. 

33. 

mdays. 

11. 

916960  dr. 

34. 

$37493.40. 

12. 

453912  gr. 

35. 

$171.60. 

13. 

67073  gr. 

36. 

$47. 2r.. 

14. 

146832^  sq.  ft. 

37. 

Wife,  $874.80; 

15. 

31556929.7  sec. 

son,  $546.75  ; 

16. 

176957  in. 

daughter,  $328.05. 

17. 

438  gi. 
81920  dr. 

38. 

$180.81. 

18. 

39. 

8  hours  ;  A,  $5.00  ; 

19. 

1518|". 

B,  $3.75 ; 

20. 

88704  in. 

C,  $2.50. 

21. 

274909^1  sec. 

40. 

$52.07. 

22. 

627264  sq.  in. 

41. 

$.97.                     a 

23. 

787^  far. 

42. 

Elder,  $12.60  ; 

24. 

4672  nv. 

younger,  $9.45. 

25. 

59405  f  gr. 

43. 

16. 

26. 

609.92  pt. 

44. 

$1113.21. 

27. 

338688  sec. 

45. 

25i. 

28. 

127.7952  dr. 

46. 

Real  estate, 

29. 

7.09038  in. 

$562.50 ; 

30. 

2342.439  sq.  ft. 

railway  shares, 

$281.25; 

city  bonds, 

Art.  162.    Page  106. 

$656.25. 

3. 

15  bu.  0  pk.  7  qt. 

47. 

Wife,  $2465  ; 

1  pt. 

eld.  son,  $1848.75 ; 

4. 

lcwt.47  1b.  15oz. 

younger  son. 

5. 

£6  16s.  10d.3far. 

$1479; 

6. 

1  111  4  5  5  3  2  3 

dau.,  $1232.50. 

7gr. 

7.  4  0.  10  f  5  4  f  3 

38  rn^. 

8.  92  rd.  1  yd.  2  ft. 

9.  81  gal.  2  qt.  0  pt. 

Igi. 

10.  £31. 

11.  3  d.  17  h.  51  min. 

55  sec. 

12.  34°  19'  9". 

13.  8  lb.  6  oz.  15  pwt. 

6gr. 

14.  228  rd.  5  yd.  1  ft. 

6  in. 

15.  2  A.  96  sq.  rd. 

4  sq.  yd. 

16.  1  wk.  2  d.  6  h.  13 

min.  20  sec. 

17.  2  cd.  47  cu.  ft. 

1502  cu.  in. 

18.  1  T.  17  cwt.  14  lb. 

13  oz.  8  dr. 

19.  5  mi.  89  rd.  1  yd. 

1  ft.  6  in. 

20.  1  sq.  rd.  1  sq.  yd. 

5  sq.  ft.  33  sq.  in. 

21.  1  mi.  33  rd.  2  yd. 

2  ft.  10  in. 

22.  2  A.  9  sq.  rd.  1  sq. 

yd.  23  sq.  ft. 

Art.  163.  Page  108. 

3.  44  gal.  3  qt.  1  pt. 

Igi. 

4.  19  cwt.  1  lb.  5  oz. 

4  dr. 

5.  164  bu.  3  pk.  7  qt. 

1  pt. 

6.  275  d.  6  h.  0  min. 

50  sec. 

7.  313°  27'  26". 

8.  80  It)  0  5  0  3  2  3 

10  gr. 


ANSWERS. 

9 

9. 

26mi.  119rd.0yd. 

15. 

19A.0sq.rd.  7sq. 

Art 

;.  167.     Page  113. 

1  ft.  6  in. 

yd.  3  sq.  ft.  9  sq. 

2. 

11  bu.  3  pk.  7  qt. 

10. 

£451  Os.  6d.  1  far. 

in. 

1  pt. 

11. 

69  T.   13  cwt.  67 

3. 

5  gal.   1  qt.  1  pt. 

lb.  10  oz.  5  dr. 

Art.  165.    Page  111. 

3gi. 

12. 

146  cd.  30  cu.  ft. 

4. 

258. 

4. 

2°  48'  45". 

1142  cu.  in. 

5. 

295. 

5. 

23  T.  9  cwt.  78  lb. 

13. 

68  lb.  9  oz.  0  pwt. 

6. 

265. 

15  oz. 

10  gr. 

7. 

695. 

6. 

25  cu.  yd.  16  cu. 

14. 

14  sq.   ml.   ij6  A. 

8. 

627. 

ft.  843  cu.  in. 

5  sq.  rd.   12  sq. 

9. 

3513. 

7. 

5  lb.  9  oz.  15  pwt. 

yd.  0  sq.  ft.  108 

10. 

7  y.  10  mo.  16  d. 

12]  gr. 

sq.  in. 

11. 

5  y.  3  mo.  20  d. 

8. 

136  rd.  4  yd.  2  ft. 

15. 

129  mi.  223  rd.  3 

12. 

1  y.  10  mo.  22  d. 

8  in. 

yd.  0  ft.  2  in. 

13. 

5  y.  8  mo.  5  d. 

9. 

£9  16s.  7d.  3i  far. 

14. 

8  y.  11  mo.  15  d. 

10. 

34  gal.  2  qt.  1  pt. 

Art.  164. 

15. 

1  y.  4  mo.  19  d. 

3gi. 

Pages  109,  110. 

11. 

6  d.  13  h.  41  min. 

3. 

10  bu.  2  pk.  6  qt. 

Art.  166.    Page  112. 

33.2  sec. 

Ipt. 

2. 

41  bu.  3  pk.  0  qt. 

12. 

4tb9§6323 

4. 

£17  19s.  Ud. 

1  pt. 

12  gr. 

2  far. 

3. 

213°  10'  50.50". 

13. 

2  mi.  45  rd.  3  yd. 

5. 

37°  46'  41". 

4. 

154  gal.  1  qt.  1  pt. 

2  ft.  10  in. 

6. 

9  gal.  3  qt.   1  pt. 

2gi. 

14. 

3  A.   132   sq.    rd. 

3gi. 

5. 

610. 15  f  5  6  f  3 

10  sq.  yd.  8  sq. 

7. 

9  lb.  11  oz.  1  pwt. 

37  m.. 

ft.  31  sq.  in. 

20  gr. 

6. 

75  mi.  144  rd.  0  yd. 

8. 

7  cu.  yd.  9  cu.  ft. 

2  ft. 

976  cu.  in. 

7. 

87  d.  6  h.  25  min. 

Art.  168.     Page  114. 

9. 

9  0.  7f  5    4  f  3 

30  sec. 

2. 

£18  2s.  M.  1  far. 

35  n^. 

8. 

£1563  2s.  Od. 

3. 

46  lb.  3  oz.  17  pwt. 

10. 

7  mi.  203  rd.  4  yd. 

3  far. 

22  gr. 

0  ft.  6  in. 

9. 

82  T.  9  cwt.  13  lb. 

4. 

5  'J\  1  cwt.  42  lb. 

11. 

14  cwt.  18  lb.  2  oz. 

6  oz.  8.5  dr. 

9  oz.  14  dr. 

Idr. 

10. 

40  cd.   64  cu.  ft. 

5. 

5  bu.  2  pk.  6  qt. 

12. 

5  A.  12  sq.  rd.  10 

987  cu.  in. 

1  pt. 

sq.  yd.  6  sq.  ft. 

11. 

253  lb.   2   oz.    10 

6. 

1  d.  23  h.  37  min. 

36  sq. in. 

pwt.  20  gr. 

15  sec. 

13. 

4  d.  19  h.  16  min. 

12. 

6  mi. 

7. 

3  mi.  20  rd.  1  yd. 

4  sec. 

13. 

56  A.  17  sq.  rd.  13 

2  ft.  10.8  in. 

14. 

16  mi.  114  rd.   5 

sq.  yd.  2  sq.  ft. 

8. 

7  gal.  2  qt.  1  pt. 

yd.  1  ft. 

66  sq.  in. 

3gi. 

10                          ACADEMIC   ARITHMETIC. 

9.  6  cd.  124  cii.   ft. 

8. 

;;  gal. 

6. 

9h.  15min.  40| 

760  cu.  in. 

9. 

>|cd. 

sec. 

10.  22°  55' 27  f.". 

10. 

4rd. 

7. 

51i.  33min.28sec. 

11.  10  lb  1  5  6  3  13 

11. 

f  cwt. 

A.M. 

5gr. 

12. 

.^0. 

8. 

91i.  5  min.  29  sec. 

Art.  169.    Page  115. 

13. 

A  A. 

A.M. 

2.  25. 

3.  31. 

4.  8. 

5.  12. 

6.  13. 

14. 
15. 

.8bu. 
.4275". 

9. 

9h.  59  min.  3^  sec. 

P.M. 

16. 

.96875  gal. 

10. 

10  h.    53  min.  57f 

17. 

.69  wk. 

sec.  P.M. 

18. 

£.148. 

11. 

89°  15'. 

19. 

.625  cu.yd. 

12. 

38°  40'  30". 

Art.  170.    Page  115. 

20. 

.8551b. 

13. 

168°  16'  15". 

3.  6  d.  3  h. 

21. 

.7532  T. 

14. 

131°  24'  30". 

4.  3  oz.  6  pwt.  16  gr. 

22. 

.3  sq.  rd. 

15. 

74°  1'  W. 

6.  2  qt.  1  pt.  3j5jgi. 

23. 

.81  mi. 

16. 

88°  12'  W. 

6.  1  pk.  1  qt.  1.2  pt. 

Art.  172. 

17. 

18°  30'  6"  E. 

7.  52'  15.6". 

Pages  117,  118. 

18. 

73°  25'  57"  W. 

8.  12  cwt.  10  lb. 

3. 

1 

19. 

72°  53'  10"  E. 

9.  6f5  5f3  20in^. 

4. 

A. 

10.  4  yd.  2  ft.  8  in. 

5. 
6. 

1  6' 

7 

Art.  174. 

11.  21  sq.yd.  1  sq.  ft. 

Pages  120-124. 

82.8  sq.  in. 

7. 

1. 

17. 

12.  3  5  4  ?i  0  3 

8. 

i. 

2. 

$97.44. 

7.68  gr. 

9. 
10. 

5 

3. 

A  train  that  runs 

13.  9s.  2d.  3 Jy  far. 

9- 
_8_ 

a  mile  in  85  sec. 

14.  58  cu.  ft.  ZU^\ 

11. 

1  1* 

4. 

Ih.  30 min.  50  sec. 

cu.  in. 

12. 

.52. 

.7. 

5. 

1  y.  8  mo.  18  d. 

15.  163  rd.  1  yd.  0  ft. 

13. 

6. 

5  A.  133sq.rd.  10 

3.6  in. 

14. 

.9616. 

sq.  yd.   0  sq.  ft. 

16.  Is.  10c?.  3.68  far. 

15. 
16. 

.75. 

108sq.  in. 

17.  91  sq.  rd.    12  sq. 

.875. 

7. 

40 rd.  1yd.  Oft.  9 

yd.  8 sq.ft.  97f 

17. 

.625. 

in. 

sq.  in. 

18. 

.27. 

8. 

5. 

19. 

.4. 

9. 

$3470.625. 

Art.  171. 

10. 

15. 

Pages  116,  117. 

Art.  173. 

11. 

$160. 

3.   |lb. 

Pages  119, 120. 

12. 

16tV. 

4.  £/,. 

3. 

3h.  13min.  48  sec. 

13. 

7. 

5.   f|°. 

4. 

6h.  48min.4sec. 

14. 

18H. 

6.  t|bu. 

5. 

llh.  54min.  241 

15. 

£4  9s.  lO^d. 

7.   Hd. 

sec. 

16. 

2023. 

ANSWERS. 

17. 

Urd.  3  yd.  1ft. 

Art.  185.    Page  133. 

25. 

.842265^1. 

18. 

$13590.72. 

3. 

361.24091. 

26. 

2.309136  sq.  rd. 

19. 

21b  115  73  03 

4. 

63.43126?. 

27. 

37.5441D1. 

2gr. 

5. 

980.203'". 

28. 

.51562i>ni. 

20. 

17tVV 

6. 

314.41948^^  "'. 

29. 

7.22624«'J  m. 

21. 

$315.15jV 

7. 

191. 94D?. 

30. 

.589932«t. 

22. 

$31.25. 

8. 

.499198'". 

31. 

128283.75CK. 

23. 

2.6. 

9. 

934.37c" '^'". 

r^R. 

34.681 5<="  dm. 

24. 

89f  cents. 

10. 

4246.567'!'. 

33. 

163.23675'ii. 

25. 

52f  ft.  per  sec. 

11. 

1.3581Hn,. 

34. 

.046206*="  Dm. 

26. 

41t)   85    03    13 

12. 

.2543772sq'". 

35. 

3.09479825Kg. 

6Jgr. 

13. 

2.26803«^?. 

36. 

2.1964347"^ 

27. 

Hh 

14. 

.2086118^'. 

37. 

2523.1344'". 

28. 

95^^jf  cents. 

15. 

.037. 

29. 

$39.69. 

16. 

.1821i>g. 

Art.  187. 

30. 

155. 

17. 

.568. 

Pages  135-137. 

31. 

8  oz.  1511  dr. 

18. 

93,6cumm. 

1. 

2.0116. 

32. 

3* 

2. 

395.4. 

33. 

8||. 

Art.  186. 

3. 

.164"^"!. 

34. 

86631  miles. 

Pages  134,  135. 

4. 

$1092. 

35. 

87°  48'  45"  W. 

3. 

9.144<i"\ 

5. 

.18162. 

36. 

$299.15. 

4. 

.88O8DI. 

6. 

.1296. 

37. 

1  of  3  gal.  2  qt.  is 

5. 

.22046  cwt. 

7. 

5.286. 

greater  by  1  gi. 

6. 

.  1308  cu.  yd. 

8. 

$39.37. 

38. 

.8228*  lb.  av. 

7. 

9.2981  dm. 

9. 

$243.8387. 

39. 

150. 

8. 

.035316  cu.  ft. 

10. 

2.679. 

40. 

123. 

9. 

.084536  gi. 

11. 

4784Dg. 

41. 

148  lb.  13.4  oz. 

10. 

10.764  sq.  ft. 

12. 

38.7. 

42. 

if 

11. 

.1772Dg. 

13. 

.03696. 

43. 

3984  mi. 

12. 

32.808  ft. 

14. 

4.047. 

44. 

$8471.771  J. 

13. 

4.7315di. 

15. 

.4971. 

45. 

8oz.l3pwt.  4Jgi'. 

14. 

3.73242HS. 

16. 

$3.77  +  . 

46. 

86f. 

15. 

1.308  cu.  yd. 

17. 

4  min.  39  sec. 

47. 

lis.  Id.  IJ  far. 

16. 

616.4288  rd. 

18. 

5.30712. 

48. 

15^. 

17. 

730.93225'-?. 

19. 

11.82875. 

49. 

.859,W5¥r- 

18. 

45.827522  cu.  in. 

20. 

$2.57  nearly. 

50. 

lUHlb. 

19. 

166.278Ha. 

21. 

134.  IH"". 

51. 

799.92  ft. 

20. 

8.213H1. 

22. 

15.55175. 

52. 

A,  2  A.  90sq.  rd.; 

21. 

.1829818  cwt. 

23. 

5.6432. 

B,  6  A.  65  sq.  rd. 

22. 

5.248176  dr. 

24. 

28.316736Kg. 

53. 

21f|- 

23. 

141.58368<=»d.n. 

25. 

9  h.  28  min. 

54. 

llOff 

24. 

81.54d8t. 

26. 

47.315i>g. 

11 


12 


ACADEMIC   ARrjHMP:TIC. 


27.   IGO.  0208  rods  a 

6. 

.3162  +  . 

8.   .854+. 

minute. 

7. 

1.4443 +  . 

9.  1.077  + . 

28.   102.30912. 

8. 

19.3864  +  . 

10.  .637  +  . 

29.  2.09+    cents  per 

9. 

26.1653  +  . 

11.  .873  +  . 

mile. 

11. 

1.6583  +  . 

12.  .912  +  . 

30.  10.35  +  . 

12. 

.4330  +  . 

13.   1.1304  +  . 

31.  6.103. 

13. 

1.2018+ . 

14.   .7862  +  . 

32.  264.17. 

14. 

.4472  +  . 

33.  32.362512. 

15. 

.6454+. 

Art.  214.    Page  150. 

34.  $35.07. 

16. 

.9045+. 

1.  33. 

35.  2.076  + Kg. 

17. 

.6236  +  . 

2.  76. 

36.  .0839+. 

18. 

.4249  +  . 

3.  88. 

37.   13.63  +  Kg. 

19. 

1.1319+. 

4.  514. 

38.  1.27  +  . 

20. 

.8552  +  . 

5.  49. 

39.  34.22  +  . 

6.  65. 

40.   13490. 7619Kni. 

Art.  212.    Page  149. 

1. 

31. 

Art.  226. 

Art.  203.    Page  143. 

2. 

4.6. 

Pages  154-156. 

1.   78. 

3. 

.88. 

4.  588sq.in. 

2.   .97. 

4. 

123. 

5.   14  sq.ft.  lOOsq.in. 

3.  21.4. 

5. 

1.14. 

6.  4isq.yd. 

4.  523. 

6. 

.098. 

7.  4  ft. 

5.  .286. 

7. 

2.02. 

8.  12  yd. 

6.  80.9. 

8. 

372. 

9.  75  ft. 

7.   .497. 

9. 

21.6. 

10.  3780  sq.  in. 

8.  .0722. 

10. 

.803. 

11.  §yd.        , 

9.  5.76. 

11. 

4.89. 

12.  75  in. 

10.  .1082. 

12. 

.317. 

13.  3|A. 

11.  21.12. 

13. 

.898. 

14.  196  sq.ft.    112  sq. 

12.   .8253. 

14. 

101.3. 

in. 

13.    900.8. 

15. 

.0534. 

15.  3U|rd. 

14.  5783. 

16. 

73.4. 

16.  32 'rd. 

15.  7.641. 

17. 

5.815. 

17.  1320. 

16.  .04738. 

18. 

.6523. 

18.  Oft. 

17.  859.35. 

19.  6600. 

18.   .98657. 

Art.  213. 

20.  630. 

Pages  149,  150. 

21.   1200. 

Art.  204.    Page  144. 

2. 

1.259  +  . 

22.  280  yd. 

2.  2.6457  +  . 

3. 

1.817  +  . 

23.  24  rd. 

3.  5.5677  +  . 

4. 

1.930+ . 

24.  163.65. 

4.  4. 1593+ . 

5. 

3.448+. 

25.  3421440. 

5.  .2828+.     ■ 

6. 

5.528  +  . 

26.  $562.50. 

ANSWERS. 


13 


27.  20  rd.  3  yd. 

28.  22  rd.  4  yd.  1  ft. 

29.  .$1878.80. 

Art.  228. 
Pages  157,  158. 

4.  25  in. 

5.  7rd.  2.iyd. 

6.  32  in. 

7.  1yd.  2  ft. 

8.  10.6066+  in. 

9.  12.2065+  in. 

10.  22  ft.  1  in. 

11.  12  ft. 

12.  60  ft.  9  in. 

13.  105.4  mi. 

14.  40.5  ft. 

15.  8  ft.  2  in. 

Art.  231. 
Pages  159,  160. 

3.  43.9824  in.; 
153.9384  sq.  in. 

4.  52.36  yd. 

5.  5.25+  rd. 

6.  13.54+  in. 

7.  24856.3392  mi. 

8.  2.67  +  . 

9.  4.507  +  . 

10.  346.3614. 

11.  61.11+  in. 

12.  110  sq.ft. 
104.5776  sq.  in. 

13.  1306.9056  sq.ft. 

14.  9075 +  . 

15.  8.862+  in. 

16.  28.3372.32  ft. 

17.  120.96  ft. 

Art.  242. 
Pages  164:-166. 

4.  330sq.  in.;  330 ca 
in. 


5.  34||  cu.  in.  ;   63| 

sq.  in. 

6.  144  sq.ft. 

7.  7  ft.;  364  sq.ft. 

8.  420  cu.  in. 

9.  392  sq.ft. 

10.  11  in. 

11.  050cu.in. 

12.  32  in. 

13.  1092  cu.  in. ;    662 

sq.  in. 

14.  3ft.;  122 sq.ft. 

15.  ^8.47. 

16.  19Hn. 

17.  2400  sq.  in. 

18.  -$55.38. 

19.  5951^  lb. 

20.  1944. 

21.  32  ft. 

22.  53|cu.  ft. 

23.  15  in. 

24.  31-lf.  [cu.in. 

25.  700sq.in.;  1568 

26.  2090880. 

Art.  248. 
Pages  169,  170. 

5.  131.9472  sq.  in.; 
197.9208  cu.  in. 

6.  201.0624  sq.  in.; 
268.0832  cu.  in. 

7.  483.8064  sq.  ft. 

8.  204.204  sq.  in. ; 
314.16  cu.in. 

9.  1847.2608  cu.  ft. 

10.  10  in.;  8  in. 

11.  10 in.;  523.6 cu.in. 

12.  8  in. 

13.  12  in. 

14.  196067256  sq.  mi.; 
258155220400   cu. 


16.  67.0208. 

16.  6  ft. 

17.  $184.07  +  . 

18.  5^1  in. 

19.  14.32  +  . 

20.  134.0416. 

21.  2.3562. 

22.  4  ft. 

23.  12  in. 

24.  6  in. 

25.  67.3698+  lb. 

26.  8.7593+. 

27.  6.109  ft. 

28.  8.169+  in. 

Art.  249. 
Pages  171, 172. 

2.  67i. 

3.  7. 

4.  65|. 

5.  552.5. 

6.  4i  ft. 

7.  5  ft. 

8.  833^  in. 

9.  65.35+  in. 

10.  48.23+. 

11.  $117. 

12.  58.5+  in. 

13.  4  ft.  7.98+  in. 

14.  $111.13  +  . 

15.  61.91+  in. 

16.  53.856+. 

17.  41.0502  +  . 

18.  4.12+  ft. 

19.  53.71  +  . 

Art.  250.    Page  173. 

2.  34. 

3.  $7.50. 

4.  48|f  ;  49. 

5.  $10.45. 

6.  $21.14j:V 


14 


ACADEMIC    ARITIIIMETIC. 


7.  28H. 

5. 

148 'lib. 

28. 

235301.94. 

8.  Across  the  room. 

6. 

7[||oz. 

29. 

37m. 

9.  651.04. 

7. 

25. 

30. 

11.2  +  Din. 

10.   With  oil-cloth. 

8. 

40. 

31. 

53.74 +m. 

9. 

432. 

Art.  251. 

10. 

54. 

MENSURATION    OF 

Pages  174, 175. 

11. 

2.72. 

SOLIDS. 

3.  $35.88. 

12 

.88. 

1. 

g^sq  dm  •    42*^"  ^'^, 

4.  $48.48. 

13. 

10.5. 

2. 

113.097689  cm. 

6.  8rV 

14. 

8.3469312. 

113.0976<="cm. 

6.  $5.40. 

3. 

549.78  sqm. 

7.  $27.53§. 

Art.  256. 

1231.5072cum. 

8.  $6. 

Pages  181-188. 

4. 

16cni;  976^1  cm. 

9.  $314.16. 

MENSURATION 

5. 

33dm. 

10.    6r. 

OI 

^    PLANE    FIGURES. 

6. 

301 .5936*1  m. 

11.  $37.71^. 

1. 

6541.5«qdm. 

7. 

4«»;  268.0832cucm. 

12.  $10.97§. 

2. 

35™. 

8. 

9.4956«qm. 

3. 

.4Hm, 

9. 

425.6c^i  Dm^ 

Art.  252. 

4. 

.0162Dm. 

10. 

9m. 

Pages  176, 177. 

5. 

144513.6ra. 

11. 

45804.528^1  mm. 

2.  lOJ. 

6. 

.094776«q  Hm. 

2061.20376cucm. 

3.  9^. 

7. 

1.83Dm. 

12. 

15m;  9m; 

4.  154^. 

8. 

13273.26^1  cm. 

1357.1712<-"m. 

5.  92|. 

9. 

3.79dm. 

13. 

407.52«im. 

6.  107if. 

10. 

1233.395«qdm. 

14. 

208.149326 ; 

7.  $17.64. 

11. 

2.5D'". 

208149.326dg. 

8.  $13,343. 

12. 

.3819+ c'". 

15. 

$13.65. 

9.  $66.85^. 

13. 

270<=m 

16. 

$199.0989. 

10.  $5,901. 

14. 

3.52782^ 

17. 

7824. 

11.  $33.68|. 

15. 

25.67566ca. 

18. 

$113.40. 

16. 

106.311741ca. 

19. 

1.38m. 

Art.  253.    Page  177. 

17. 

65.9m. 

20. 

1.5372. 

2.  8.484  in.; 

108. 

18. 

87.9Dm. 

21. 

48.55344KK. 

3.   11.312  in.; 

224. 

19. 

221m. 

22. 

5309.304. 

4.  10.605  in.; 

162  ^ 

20. 

5500. 

23. 

638.4. 

5.  9.898  in.; 

128 -. 

21. 

$6653.36. 

24. 

37000. 

.  6.  13.433 in.; 

358/,V 

22. 

65.1Kni. 

25. 

19373.2. 

23. 

10.23250536. 

26. 

3650.5392Dg. 

Art.  254. 

24. 

1.6493925. 

27. 

13.8984384Kg. 

Pages  178, 

,179. 

25. 

11.5m. 

28. 

579.63588. 

3.  550.6251b 

26. 

5.939 +  Hm. 

29. 

.33m. 

4.  3037.51b. 

27. 

7.6Dni. 

30. 

3.8050012. 

ANSWERS. 

CAPACITY       OF      BINS, 

11.   11568.72d8. 

10 

.  49^. 

TANKS,      AND      CIS- 

12.   65cni. 

11 

.  $331.25. 

TERNS,  CARPETING, 

13.  7.1. 

12 

.  $1125. 

PLASTERING,       AND 

14.  2.08. 

13 

,  280. 

PAPERING. 

15.    9dm. 

14. 

,  $20.40. 

1.  5801.6. 

16.  20.5633428Kg. 

15. 

159  ft.  41  in. 

2.  49.68. 

3.  46.13. 

4.  $27.63. 

Art.  263. 
Pages  191,  192. 

1.  2:5. 

2.  8 :  15. 

16. 
17. 
18. 

8|. 

42  mi.  an  hour. 

$29.80. 

5.   1.92"'. 

19. 

8^- 

6.   26.9dn\ 

3.  23:37. 

4.  9:10. 

5.  9:13. 

6.  22:15. 

7.  11:14. 

8.  5:3. 

9.  7:9. 

10.  25  :  27. 

11.  8:5. 

12.  2  :  3. 

13.  16:25. 

14.  64:45. 

20. 

10  min.  30  sec. 

7.   134.3034; 

21. 

2Jh. 

134303.4Hg. 

22. 

51. 

8.  6.5. 

9.  1.7'". 

23. 
24. 

276  lb.  32  oz. 
105  d. 

10.  $471.24. 

25. 

$151t. 

11.  $60.42. 

12.  11  h.  30  mm. 

Art.  273. 

13.  10.31. 

Pages  198-201. 

14.  15.927912. 

15.  3.5'». 

16.  $40,026. 

17.  13.5. 

2. 
3. 
4. 
5. 

135. 

8. 
20. 

18.  Across  the  room. 

Art.  270.    Page  194. 

6. 

46i. 

19.  $102,971; 

3.    112. 

7. 

5. 

3l07.852Kg. 

4.  36. 

8. 

12. 

20.  1.8'». 

5.  24. 

9. 

2f.^ 

21.  $34,104  ;  $34,104. 

6.  If. 

10. 

9. 

22.   75cn'. 

7.  i-|. 

11. 

216  mi. 

23.   .997+"'. 

2  5 

8.  -V-. 

12. 

42i. 

24.  $7.98. 

9.  1§. 

13. 

■Sj%  oz. 

25.   1.5"'. 

10.  V. 

14. 

^■ 

15. 

72. 

Art.  257. 

Art.  271. 

16. 

660  lb. 

Pages  189,  190. 

Pages  196,  197. 

17. 

10. 

4.  12008. 

3.  $4.51. 

18. 

m  ft. 

5.  1317Hg. 

4.  $15.96. 

19. 

7^. 

6.   18.7. 

6.  511. 

20. 

78|  lb. 

7.  12.9. 

6.  65^. 

21. 

13|. 

8.  .92. 

7.  648f. 

22. 

Oft. 

9.   13.596. 

8.  78f 

23. 

4. 

10.  2.30679. 

9.  $4.60. 

24. 

4^  ft. 

15 


16 


ACADEMIC    ARITHMETIC. 


25.  20}. 

9. 

4  ft.  ()  in. 

4. 

A,  $26.40; 

26.  1150  1b. 

10. 

9  in. 

B,  $13.20; 

27.  28. 

11. 

$37.80. 

C,  $33  ;  D,  $44. 

28.  81. 

12. 

218  lb.  12  oz. 

5. 

Adams,  $1200; 

29.  8. 

13. 

7.93 -f  in. 

Burke,  $1120. 

30.  5,^ 

14. 

2  h.  6  min. 

6. 

A,  $1275; 

15. 

1  ft.  4  in. 

B,  $1020; 

Art.  275. 

C,  $1360. 

Pages  202,  203. 

Art.  280. 

7. 

Hand,  $704 ; 

3.    14,  35,  56. 

Pages  206,  207. 

Sears,  $720 ; 

4.  140,  184. 

2. 

A,  $600;  B,  $860. 

Thomas,  $768. 

5.  39,05,91,117,143. 

3. 

Allen,  $1250  ; 

8. 

A,  $532  ; 

6.  672,  630,  588,  576. 

Brown,  $1500  ; 

B,  $148. 

7.  50|,  355}. 

Cole,  $800. 

9. 

A,  $48.75; 

8.  510  lb.  copper. 

4. 

A,  $192.75; 

B,  $39;  C,  $22.75. 

306  lb.  zinc. 

B,  $64.25. 

10. 

P^uller,  $200  ; 

119  lb.  tin. 

5. 

A,  $49.24  ; 

Gray,  $250. 

9.  A,  .$375  ;  B,  $480  ; 

B,  $61.55; 

11. 

A,  $145; 

C,  $612. 

C,  $73.86. 

B,  $185 ; 

10.  182f  parts  salt- 

6. 

A,  $17120; 

C,  $200. 

petre, 

B,  $8560  ; 

12. 

A,  $1500; 

34}  parts  charcoal, 

C,  $1712. 

B,  $1800  ; 

57Y\r  parts  sul- 

7. 

Hale,  $1725  ; 

C,  $3375. 

phur. 

Hunt,  $2070. 

13. 

Lowe,  $2035 ; 

11.  42,  84,  252.  1008. 

8. 

A,  $1077.36 ; 

Martin,  $2255; 

12.  $23.36,  $35.04, 

B,  $1258.02  ; 

Neal,  $776. 

$43.80,  $01.32. 

C,  $3145.05. 

13.  26 II  cu.  ft.   oxy- 

9. 

A,  $5.25  ; 

Art.  287.     Page  210. 

gen,  99|^  cu.  ft. 

B,  $3.75  ; 

8. 

.056. 

nitrogen. 

C,  $6.00  ; 

9. 

.008. 

14.  30,  20,  16. 

D,  $4.50. 

10. 

.001875. 

15.  $115. 

10. 

A,  $867.84; 

11. 

.0041. 

16.  15,  30,  55,  90. 

B,  .$578.56; 

C,  $1084.80. 

Art.  288.     Page  211. 

Art.  276. 

1. 

if- 

Pages  204,  205. 

Art.  282. 

2. 

I' 

3.  540  sq.  in. 

Pages  208,  209. 

3. 

iV- 

4.  8  in. 

2. 

A,  $162 ; 

4. 

tV 

5.  708J  cu.  in. 

B,  $194.40. 

5. 

hh 

6.  6  ft. 

3. 

A,  $82.50  ; 

6. 

If 

7.  1  ft.  4|  in. 

B,  $247.50  ; 

7. 

8 

8.  7  in. 

C,  $137.50. 

8. 

jh' 

ANSWERS. 

17 

9.  tU- 

13. 

tVo- 

24. 

$  10.38.54  j-V 

10.  tI^- 

14. 

3  ft.  6^  in. 

26. 

f- 

11-  ?VV 

16. 

$413. 

26. 

.$45.75. 

12.   3^. 

16. 

$415. 

27. 

146ft.  Sin. 

13.  ^',. 

17. 

$953.04. 

28. 

7608. 

14.  ^h- 

18. 

.$498.27. 

29. 

$1286.40. 

15-  Th- 

19. 

$1132.25. 

30. 

$107.80. 

20. 

$378.42. 

31. 

2928. 

Art.  289.     Page  211. 

21. 

$2257.57. 

32. 

135  girls,  162  boys. 

2.  57^%. 

22. 

14093. 

33. 

$17500. 

3.  1H%. 

23. 

$891. 

34. 

1241,  816. 

4.   n2i%. 

24. 

.$9.45. 

35. 

$2220.35. 

5.  58J%. 

25. 

$.03. 

36. 

$5688. 

6.  85%. 

26. 

10401b.  lead; 

37. 

$1000.50. 

7.   16%. 

16  lb.  silver. 

38. 

$1015.30. 

8.  7^/0. 

27. 

$1815. 

39. 

57615. 

9.  131io/„. 

28. 

$344.75. 

40. 

$1250. 

10.  73.^0/^. 

29. 

703. 

11.  171^0/,. 

12.  54p/o. 

30. 

lin. 

Art.  292. 

31. 

$.46^0. 

Pages  219-221. 

13.  fi%. 

32. 

$1573. 

4. 

6,1%. 

14.  1%. 

5. 

40%. 

15.  if  %. 

16.  f%. 

Art.  291. 

6. 

54^/0. 

Pages  215-218. 

7. 

74%. 

17.  f%. 

3. 

.32.4. 

8. 

9ir/o. 

18.   123i\%. 

4. 

$18.50. 

9. 

75%. 

19.  l3\%. 

6. 

¥• 

10. 

108% 

20.  2l5V%. 

6. 

17°  45'. 

11. 

63«%. 

21.  49||o/o. 

7. 

2^,. 

12. 

31i%. 

8. 

47  ft.  8  in. 

13. 

1%. 

Art.  290. 

9. 

W- 

14. 

42f%. 

Pages  212-214. 

10. 

3. 

16. 

i%. 

3.  4.291. 

11. 

¥• 

16. 

4ir/o. 

4.  .17.77. 

12. 

187  lb.  4  oz. 

17. 

12]%. 

5.  62  bu. 

13. 

2^f. 

18. 

76%  silver;   24% 

6.  ^,3.. 

14. 

$193.75. 

copper;  31^%. 

7.  f. 

18. 

$83.50. 

19. 

19|%- 

8.   1. 

19. 

$1328. 

20. 

26tr/o. 

9.  .$2.27. 

20. 

384. 

21. 

221%. 

10.  tI^. 

21. 

656.25. 

22. 

181%. 

11.  £100  2^, 

22. 

$425. 

23. 

87^%. 

X2.  f-, 

23. 

$62.50. 

24. 

20%. 

18 


ACADEMIC    ARITHMETIC. 


25.   1%. 

7. 

$520;  $31.20. 

3. 

,  $17.50  on  $1000. 

26.  2.i%. 

8. 

$990,099+  ; 

4. 

$54.28. 

27.  m%- 

$9,901. 

5. 

$4200. 

28.  28%. 

9. 

3^  %. 

6. 

$7000. 

29.  170/0. 

10. 

400. 

7. 

$215.79. 

30.  l()f%. 

11. 

$2496. 

8. 

$12.75  on  $1000; 

31.   19%. 

12. 

$7840. 

$107.10. 

32.  8.9%. 

13. 

$2364.18. 

9. 

$35700. 

33.  6|%;  61%. 

14. 

$438  18. 

10. 

$332. 

34.  15^%. 

15. 

$2444.98+  ; 

11. 

$11.30  on  $1000. 

35.  2.^0/^. 

$55.02. 

12. 

$16  on  $1000. 

36.  19i%. 

16. 

$8400. 

13. 

$22401. 

37.  37^  %. 

17. 

.$2157.40. 

18. 

336. 

Art.  303. 

Art.  294. 

19. 

$632. 

Pages  232,  233. 

Pages  222-224. 

20. 

$6390. 

3. 

$7.98. 

3.  $45.75. 

21. 

3.50. 

4. 

$100.36. 

4.  $203.28. 

22. 

13  ;  $8.84. 

5. 

$157.6746. 

5.  .|14.70. 

23. 

96. 

6. 

$151.20. 

6.  $im. 

7. 

$82.40. 

7.  18.00. 

Art.  298. 

8. 

$61.11. 

8.  .S4..35. 

Pages  228,  229. 

9. 

$1614.60. 

9.  {$14.28. 

3. 

$47.97. 

10. 

$690. 

10.  Loses  3|  %. 

4. 

$415.25. 

11. 

55%. 

11.  $1.26. 

5. 

$1860. 

12. 

$77.50494. 

12.  $2653.56. 

6. 

h%- 

13. 

$293.443375. 

13.  .$2.10. 

7. 

$1720. 

14. 

$904.8553125. 

14.  9fo/„. 

8. 

$3475.50. 

15.  $24.75. 

9. 

$2009.60. 

Art.  307. 

16.  $13.32. 

10. 

-1%. 

Pages  235,  236. 

18.  33|o/,. 

11. 

$5760. 

4. 

$217.50. 

19.  50%. 

12. 

$9600. 

5. 

$33.78. 

20.  27|o/o. 

13. 

$2625. 

6. 

$16.96. 

21.  36i%. 

14. 

$2653.75. 

7. 

$4.74. 

22.37^0/0. 

15. 

$108.50. 

8. 

$108.35. 

16. 

$163.50. 

9. 

$2.67. 

Art.  296. 

17. 

$7980. 

10. 

$21.28. 

Pages  225-227. 

18. 

3|%. 

11. 

$14.11. 

3.  $18.72. 

12. 

$50.36. 

4.  $4964.82. 

Art.  301. 

13. 

$18.08. 

5.  50/0. 

Pages  230,  231. 

14. 

$2.61. 

6.  $21.25. 

2. 

$13,50  ou  $1000, 

15. 

$14.44. 

ANSWERS. 

19 

16.  $197.95. 

10. 

$197.45. 

Art.  312.    Page  244^ 

17.  $76.39. 

11. 

$845.54. 

3.  $310. 

18.  $276.37. 

4.  $107.75. 

19.  $2214.36. 

Art.  310. 

6.  $196. 

20.  $467.99. 

Pages  240,  241. 

6.  $297.20. 

21.  $3794.82. 

2. 

21%. 

7.  $178.80. 

22.  $1102.58. 

3. 

31%. 

8.  $657.16. 

23.  $636.46. 

4. 

6%. 

9.  $86.34. 

5. 

2%. 

10.  $947.93. 

Art.  308. 
Pages  238,  239. 

6. 

H%- 

11.  $74.16. 

4.  $161.35. 
6.  $10.33. 

6.  $6.42. 

7.  $1.05. 

8.  $8.12. 

9.  $7.08. 

10.  $86.68. 

11.  $20.17. 

7. 
8. 
9. 

2r/o. 

7%. 

1|%- 

12.  $305.48. 

13.  $.325.85. 
14-  $250.91. 

10. 
11. 
12. 

2r/o. 

4%. 

3%. 

16.  $342.86. 

16.  $40.74. 

17.  $573.25. 

13. 
14. 

4J%. 

18.  $4191.04. 

19.  $589.66. 

12.  $36.80. 

15. 

8%- 

20.  $1.36.08. 

13.  $333.60. 

16. 
17. 

n%. 
3%. 

Art.  314.    Page  246. 

14.  $5183.46. 

2.  $188.24. 

15.  $1047.72. 

Art   .-^11 

3.  $27.21. 

17.  $162.17. 

Pages  242,  243. 

4.  $31.06. 

18.  $13.65. 

19.  $1.24. 

20.  $24.25. 

21.  $38.23. 

22.  $1.73. 

23.  $142.06. 

3. 

•     4. 

6. 

6. 

7. 

4  y.  3  mo. 

2  y.  10  mo. 
6  mo. 

3  y.  1  mo. 

5  mo.  24  d. 

5.  $103.02. 

6.  $52.88. 

7.  -$24.38. 

8.  $37.95. 

9.  $38.43. 

24.  $157.11. 

8. 

11  mo.  12  d. 

Art.  323. 

25.  $9179.74. 

9. 

1  y.  8  mo.  12  d. 

Pages  249,  250. 

26.  $328.47. 

10. 

6  mo.  3  d. 

2.  $318.35. 

11. 

9  mo.  6  d. 

3.  $244.24. 

Art.  309.   Page  240. 

12. 

4  mo.  7  d. 

4.  $606.91. 

2.  $921.18. 

13. 

2  mo.  11  d. 

5.  $90.66. 

3.  $1.26. 

14. 

8  mo. 

6.  $371.65. 

4.  $19.38. 

15. 

21  d. 

7.  $183.05. 

6.  $62.59. 

16. 

9  mo.  29  d. 

8.  $521.25. 

6.  $10.53. 

17. 

5  y.  5  mo.  0  d. 

7.  $1.92. 

18. 

7  mo.  12  d. 

Art.  324. 

8.  $6689.71. 

19. 

16  y.  8  mo. 

Pages  251,  252. 

9.  $250.61. 

20. 

22  y.  2  mo.  20  d. 

2.  $309.85. 

20 


ACADEMIC   ARITHMETIC. 


3.  $251.39. 

4.  !$754.70. 
6.  $766.71. 

6.  $244.06. 

7.  $436.79. 

8.  $327.70. 

9.  $345.68. 

Art.  326. 
Pages  253,  254. 

2.  $156.63. 

3.  $258.81. 

4.  $59.85. 
6.  $167.24. 

6.  $464.16. 

7.  $180.89. 

8.  $801.87. 

9.  $624.48. 

10.  $143.42. 

11.  $1790.82. 

Art.  327.   Page  255. 

2.  $566.34. 

3.  $1696.18. 

4.  $429.25. 
6.  $639.00. 

6.  $788.13. 

7.  $696.71. 

8.  $771.80. 

Art.  328.  Page  257. 

2.  $133.49. 

3.  $1562.79. 

4.  $422.37. 

5.  $272.07. 

6.  $265.16. 

7.  $298.30. 

Art.  329.  Page  258. 

2.  $1059.62. 

3.  $469.21. 

4.  $96  86. 

5.  $1016.86. 


6.  $1196.76. 

Art.  331.    Page  260. 

2.  $340.43  ;  $59.57. 

3.  $868.29;  $21.71. 

4.  $595.47  ;  $129.53. 

5.  $1549.25; 
$180.75. 

6.  $662.46;  $19.54. 

7.  $267.46;  $1.74. 

8.  $2454.19;  $45.81. 

9.  $907.50;  $42.50. 

10.  $127.07  ;  $8.68. 

11.  $342.21  ;  $5.47. 

Art.  334. 
Pages  261-263. 

3.  $492.25. 

4.  $941.69. 

5.  $36.17. 

6.  $5.58. 

7.  $996.33. 

8.  $5.35. 

9.  $6009.95. 

10.  $420.05. 

11.  $802.98. 

12.  $3.33. 

13.  $6.43. 

14.  $2980.75. 

15.  $191.44. 

16.  $1.13. 

17.  $1128.73. 

18.  $199.88. 

19.  $4.67. 

20.  $279.76. 

21.  $396.59. 

22.  $3.71. 

Art.  335.  Page  264. 

2.  $609.45. 

3.  $341.57. 

4.  $227.39. 
6.  $8245.30. 


6.  $554.81. 

7.  $1535.12. 

8.  $430.27. 

9.  $921.80. 
10.  $377.48. 

Art.  343. 
Pages  267,  268. 

3.  $506.25. 

4.  $278.60. 

5.  $7874. 

6.  $474.56. 

7.  $1928.16. 

8.  $693.61. 

9.  $344.97. 

10.  $606.80. 

11.  $1333.59. 

Art.  344. 
Pages  260,  270. 

3.  $440. 

4.  $649.60. 

5.  $918.46. 

6.  $241.81. 

7.  $752.10. 

8.  $192. 

9.  $584.65. 

10.  $2985.07. 

11.  $2346.71. 

Art.  348. 
Pages  272,  273. 

4.  $1693.89. 

5.  $201.93. 

6.  $581.40. 

7.  £39  4s. 

8.  29942.55  fcs. 

9.  $78.89375. 

10.  $515.20. 

11.  $139.83. 

12.  29368  mks. 

13.  $2557.697625. 

14.  £176. 


ANSWERS. 


21 


15.  4914.70125  fcs. 

16.  $466.4109375. 

17.  £132  8s.  9d. 

18.  1501.03  mks. 

19.  .$5248.19. 

Art.  350. 
Pages  275,  276. 

4.  7  mo.  10  d. 

5.  8  mo.  27  d. 

6.  73  d. 

7.  April  23. 

8.  Oct.  10. 

9.  Nov.  18. 

10.  Jan.  8,  1890. 

11.  June  5,  1891. 

12.  Dec.  24. 

13.  July  29,  1892. 

14.  After  9  mo.  15  d. 

15.  Oct.  2. 

16.  5  mo.  27  d.  after 

it  becomes  due. 

17.  Oct.  14, 1892. 

Art.  352. 
Pages  279-281. 

2.  Nov.  12. 

3.  Jan.  11,  1891. 

4.  Feb.  13. 

5.  April  26. 

6.  Nov.  5,  1890. 

7.  Dec.  2. 

8.  Dec.  27,  1890. 

Art.  363. 
Pages  286-291. 

5.  $17741.25. 

6.  $8011.50. 

7.  $5466.75. 

8.  $2588.25. 

9.  $1354.50. 
10.  $4977. 


11.  46. 

12.  $7500. 

13.  43. 

14.  626. 

15.  $15000. 

16.  75. 

17.  153. 

18.  169|. 

19.  8^o/o. 

20.  80|. 

21.  12|%. 

22.  36|%. 

23.  186. 

24.  $395.50, 

25.  $75000. 

26.  $3982.75. 

27.  1|%. 

28.  $7900. 

29.  38. 

30.  153. 

31.  375. 

36.  $160. 

37.  $150. 

38.  $271.25. 

39.  $484.50. 

40.  $4431. 

41.  $5797.50. 

42.  $12060, 

43.  $7449.75. 

44.  5%. 

45.  4^o/o. 

46.  6.4%. 

47.  Sh%- 

48.  224. 

49.  74. 

50.  146|. 

51.  67|. 

52.  150i. 

53.  A  4|%  stock  at 

90. 

54.  Increased  $11. 
65.  61^. 


66.  The    investments 

are  equally  good. 

67.  Diminished 

$23.50. 

58.  $6757.50  invested 

in  6%  bonds  at 
112f. 

59.  4io/„. 

60.  A  3 1  %  stock  at  a 

disc't  of  28 1  %. 

61.  129. 

62.  Increased  $36.75. 

Art.  367. 
Pages  293-295. 

2.  72. 

3.  32^. 

4.  85. 

5.  51.6. 

6.  V-- 

7.  70,  444. 

8.  289,  3129. 

9.  69,  5249. 

10.  31,  9306. 

11.  81|,  2040. 

12.  9.9,  748.3. 

13.  V-,  -Hl^- 

14.  h  W- 

15.  4950. 

16.  2550. 

17.  The  last  term  is 

18.  1197. 

19.  44550, 

20.  498^2    ft.,    4117^ 

ft. 

21.  15f  mi.,  325r\mi. 

22.  $3850. 

Art.  372. 
Pages  297,  298. 

3.  2500. 


22 


ACADEMIC   ARITHMETIC. 


4.  W- 

$1780.73. 

31.  123-1. 

6.  ^%- 

7.  $280.90. 

32.  $1466.9424. 

6.    z\'-2' 

8.  $351.52. 

33.  $3151.82. 

7.  1024,2047. 

9.  $169.21. 

34.  11%. 

8.  13122,  19680. 

35.  564245  in. 

9-  jh>  -\W- 

Art.  377. 

36.  5  h.  46ff  min. 

10.  ,V¥^,  WiV- 

Pages  303-317. 

37.  .$447.15;  $31.80. 

11.  -W-.  ¥#-. 

1.  4936095. 

38.  2088.77  mks. 

12-  /3»  HF. 

2.  9Mf. 

39.  1,VV 

13.  H-gJ-. 

3.  VtVW-.      . 

40.  4420  sq.  in.  ; 

14.  HIF- 

4.  $39.20. 

18928  cu.  in. 

15.  $163.83. 

5.  $602.98. 

41.  $21356.25. 

16.  1  mi.,  767}  mi. 

6.  $570.42. 

42.  $738.92. 

17.  The  last  term  is  f . 

7.   19  rd.  5  yd.  1  ft. 

43.   1008. 

18.  17576. 

8.  i. 

44.  if. 

19.  $121.550625. 

9.   Hind-wheel,  5580; 

45.  4032. 

fore-wheel,  6600. 

46.  $11059.58. 

Art.  373.   Page  299. 

10.  .853976. 

48.  .0875. 

3.  $7203.26. 

11.  $149.60. 

49.  .$54.81  ;  $54.23. 

4.  $9733.22. 

12.  198  mi.  195  rd. 

50.  3|i. 

5.  $78.74. 

2  yd.  Oft.  9  in. 

51.  $987.65;  $12.35. 

6.  $132.42. 

13.   122"  27'  17"  W. 

52.  523. 

7.  $24000. 

14.  m- 

63.  5^%. 

8.  $4800. 

15.  3i-f 

54.  fli. 

9.  $992. 

16.  U- 

55.  .$34.79. 

17.  6876. 

56.  2.375. 

Art.  375.    Page  301. 

18.  $1500. 

57.  $2801.25. 

4.  $1620  ;  $1350. 

19.  6  y.  3  mo.  12  d. 

20.  .73142857. 

58.  l|f-f. 

59.  7.6985. 

5.  $783.75  ;  $690.53. 

6.  $7520;  $5371.43. 

7.  $5607;  $4441.19. 

8.  $408. 

9.  $620. 
10.  $272. 

21.  11.312  in.  ;  208|. 

22.  49.34if. 

23.  18  in. 

60.   A,  $248.82  ; 

B,  $317.46  ; 

C,  $197.34; 

24.  ^nh- 

D,  $351.78. 

25.  .$273.75. 

61.  23. 

26.  32f%. 

27.  18744264. 

62.  3.36  in. 

11.  $642. 

63.  10  ft.  1  in. 

28.  1  sq.  mi.  468  A.  96 

64.  2  mi.  159  rd.  3  yd 

Art.  376.    Page  302. 

sq.  rd.  7  sq.  yd. 

Hft. 

4.  $630.50  ;  $544.65. 

8  sq.  ft.  29  sq.  in. 

65.  304. 

5.  $1312.38; 

29.  .$253.50. 

66.  4ff|;  A,   .$25.20 

$1039.53. 

30.  6  h.  37  min.  30| 

B,  $21  ;  C,  $18 

6.  $2166.53; 

sec.  P.M. 

D,  $15.75. 

ANSWERS. 


23 


67.  $1472.06, 

68.  2  d.  8  h.  18  min. 

45  sec. 

69.  $19.60. 

70.  9  oz.  11  pwt. 
14.7264  gr. 

71.  85J. 

72.  9J3  ;  2991^. 

73.  tVo- 

74.  68ii;  201  Hi. 

75.  74  sq.  mi.  :]80  A. 

102  sq.  rd.  14  sq. 
yd.  3  sq.  ft.  61 
sq.  in. 

76.  $9185.75. 

77.  £4  9s.  Id.  I  far. 

78.  A,  $39.50; 

B,  $38.71; 

C,  $52.14. 

79.  4  ft.  fOgV  in. 

80.  .06444625. 

81.  $874.80. 

82.  $2906.14. 

83.  lOii. 

84.  3|. 

85.  $949.62. 

86.  $1744.688915. 

87.  if  mi. 

88.  Wife,  $3150; 
son,  $2940; 
dauglit'r,$2866.50. 

89.  826. 

90.  70  d. 

91.  ft,  Vih  f^. 

92.  11  y.  9  mo.  21  d. 

93.  12  d.  14  h.  0  min. 

25  sec. 

94.  26r«o\- 

95.  174|. 

96.  $53.90. 

97.  9986J|mi. 

98.  2800733. 


99.  19.584. 

100.  $118070.75. 

101.  81^Y3- 

102.  8010,  12460, 
17355,  22428, 
27590. 

103.  $5038.20; 
$3650.87. 

104.  .860855. 

105.  $6.44. 

106.  IxV/i- 

107.  21ft.  9.629+ in. 

108.  479332^  gr. 

109.  $1040.72. 

110.  40,^^. 

111.  £88  16s.  7c?.  3 far. 

112.  May  20. 

113.  $547.18. 

114.  271%. 

115.  $861.69. 

116.  31b.2oz.5.376dr. 

117.  2_43^2^ 

118.  $67.31. 

119.  $863.95. 

120.  2  ft.  8  in. 

121.  /o- 

122.  $.03J. 

123.  ^\\h- 

124.  $13.75  on  $1000; 
$122.75. 

125.  4.753+ in. 

126.  $2376.36. 

127.  37J3. 

128.  393  A.  135sq.rd. 
11  sq.  yd.  5  sq.ft. 
102^*^3  sq.  in. 

129.  $389.91. 

130.  209  rV 

131.  16794.50112     cu. 

in. 

132.  .8796  +  . 

133.  605  ft. 


134.  .897216796875  1b. 

135.  7  lb.  1  oz.  3  pwt. 

6||gr. 

136.  $179.96. 

137.  ,\V 

138.  .0738  cd. 

139.  26880. 

140.  ^H!^. 

141.  3,Vo/^;  Zn. 

142.  $25709.25. 

143.  IHfini-; 

144.  175.76. 

145.  7  mo.  14  d.   after 

it  becomes  due. 

146.  29^0/^. 

147.  19  T.   12  cwt.  91 

lb.5oz.  12ifdr. 

148.  4  ft.  6.2+ in. 

149.  $1093.09. 

150.  4^V 

151.  23.7. 

152.  190.5904. 

153.  19|. 

154.  $21968. 

155.  A,  $449.75; 

B,  $428.75; 

C,  $458.50. 

156.  $582. 

157.  $668.91. 

158.  2~jV  rd. 

159.  8028979200  sq.in. 

160.  iV,. 

161.  \\K 

162.  8i%. 

163.  $2155.06; 
$1772.98. 

164.  2/^. 

165.  Gained  $15.78, 

166.  .71875. 

167.  1|. 

168.  287.354-  lb. 


24 


ACADEMIC    ARITHMETIC. 


169.  .1335.95. 

170.  Diminished 

$179.25. 

171.  1954.3265281b. 

172.  IJ. 

173.  3  h.  19141  min. 

174.  13120|. 

175.  .1546.09. 

176.  3128.4. 

177.  14|. 

178.  1,1^. 

179.  $41.94|;  $11.67^. 

180.  $218.35. 

181.  March  7,  1890. 

Art.  378. 
Pages  317-320. 

1.  .527^" cm;  .0527^8. 

2.  965.7cumm. 

3.  .34250™. 

4.  $49.83. 

6.  9.6558Hm  a  min. 

6.  141.696Dg. 

7.  4..3443906Km. 

8.  $156.77  +  . 

9.  112.373125^1. 

10.  4.97096. 

11.  2.01168. 

12.  2.674224a. 

13.  230  gal.  3  qt.  1  pt. 

.30656  gi. ; 
24  bu.  3  pk.  1  qt. 
.90432  pt. 

14.  1.491 +  c.  per  kilo- 

meter. 

15.  226.0737  ; 

2260.737. 

16.  562500. 

17.  $95,496. 

18.  4158. 

19.  $1542.03. 

20.  2.6753T. 

21.  .$40.08;  $39.42. 


22.  .5899H™. 

23.  53.0079c"  d™. 

24.  96.19830528. 

25.  $35.01485526. 

26.  .178Dtn. 

27.  964.5. 

28.  18dm. 

29.  23  A.  28  sq.  rd.  14 

sq.  yd.  3  sq.  ft. 
116.4672  sq.  in. 

30.  2.2763736. 

31.  109.1674584Hg. 

32.  590.64552Dg. 

33.  5822.4075. 

34.  9.0792248qdm, 

35.  .1432 -}-m. 

36.  .175D«i. 

37.  .21m; 
.00494802'"!  Dm. 

38.  1396mm. 

39.  201.0624. 

40.  2.57. 

41.  8  lb.  5  oz. 

5.347328  dr.  ; 
10  lb.  1  oz.  10  pwt. 
12.96  gr. 

42.  655.02+  lb. 

43.  62566400. 

44.  72.572  +  . 

45.  3  h.  19  min. 

46.  $343308.42. 

47.  156.0872544. 

Appendix. 
Page  324. 

4.  25°  C. ;  20°  R. 

5.  60°  C. ;  48°  R. 

6.  -13\°C.; 
-105°  R. 

7.  -36S°C. ; 
-29^°  R. 

8.  131°  F. ;  44°  R. 

9.  158°  F.  ;  56°  R. 


10.  10|°F.  ;   -9f°R. 

11.  -13°  F. ; 
-20°  R. 

12.  149°  F. ;  65°  C. 

13.  88J°F.;  31^°  C. 

14.  9^°F.;   -12^°  C. 

15.  -17^°  F.; 
-27^°  C. 

Page  332. 

2.  £7  14s.  m.  2.5 

far. ; 
£63   19s.    2c?.  2.5 
far. 

3.  18s.  2d.  1.072  far.; 
£32  12s.  8d 

1.072  far. 

4.  15s.  IM.  3.65  far.; 
£28  4s.  M. 

3.55  far. 

5.  £2  Os.   lid.    2.05 

far. ;  £43  Os.  Id. 
3.05  far. 

6.  6io/„. 

7.  2  y.  1  mo.  15  d. 

8.  £14  1.3s.  9d. 

Page  339. 

7.  10010111. 

8.  114144. 

9.  1576^2. 

10.  100120100112; 
557663;  90e?e. 

11.  3230. 

12.  1826. 

13.  16046. 

14.  51692. 

15.  100000000000. 

16.  547771. 

17.  1433423. 

18.  1032e. 

19.  13122^5. 

20.  8536. 


THIS  BOOK  IS  DUE  ON  THE  T.AST  DATE 
STAMPED  BELOW 

AN  INITIAL  FINE  OF  25  CENTS 

WILL   BE  ASSESSED    FOR   FAILURE  TO    RETURN 
THIS    BOOK   ON    THE   DATE   DUE.    THE   PENALTY 
WILL  INCREASE  TO  50  CENTS  ON  THE  FOURTH 
DAY    AND    TO     $1.00    ON     THE    SEVENTH     DAY 
OVERDUE. 

Stl-   2-  ^^^ 

i)'"" 

UL 

LD  21-100m-8,'34 

'  O     I    /H#C> 


V 


